Step 1: We consider the unbounded operators \(a(f)\)  and \(a^*(f)\)  on
\(S(\mathcal {H})_0 \subseteq S(\mathcal {H})\)  and observe that \(\)  is the common kernel
of the annihilation operators \(a(f)\) . For \(g\in \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) ,
we observe that \(\pi _s(g)\Omega \)  lies in the common kernel of the
operators \(a_g(f) = \pi _s(g) a(f) \pi _s(g)^{-1}\) 
(cf. Remark REF ).
If \(g_1\)  is invertible, which is the case in some open
\(\mathbf {1}\) -neighborhood in \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\)  (actually on the whole group),
we consider the antilinear operator \(T(g) := g_2 g_1^{-1}\) 
for which \(a_g(g_1^{-1} f) = a(f) + a^*(T(g)f)\) .
Therefore \(F = \pi _s(g)\Omega \)  is a solution of the
following system of equations:
\(a(f)F = - a^*(T(g)f)F \quad \mbox{ for } \quad f \in \mathcal {H}.\) 
Step 2: For each \(n \in {\mathbb {N}}\) , the subset \(a^*(\mathcal {H})S^n(\mathcal {H})\)  is total
in \(S^{n+1}(\mathcal {H})\) , which implies that
\(\lbrace  T \in S^{n+1}(\mathcal {H}) \colon  (\forall f \in \mathcal {H})\, a(f)T = 0\rbrace = \lbrace 0\rbrace .\) 
If an element \(F= \sum _{n = 0}^\infty F_n \in S(\mathcal {H})\) 
with \(F_n \in S^n(\mathcal {H})\)  satisfies
(REF ), then
\(a(f)F_1 = 0, \quad a(f)F_{n+1} = - a^*(T(g)f)F_{n-1} \quad \mbox{ for } \quad f \in \mathcal {H},n \in {\mathbb {N}}.\) 
This implies \(F_1 = 0\) , and inductively we obtain with
(REF ) \(F_{2k+1} = 0\)  for \(k \in {\mathbb {N}}_0\) .
We also derive from (REF ) that \(F\)  is completely determined
by \(F_0\) , hence that the solution space of (REF )
is at most one-dimensional.
If \(F\)  is a solution, we may w.l.o.g. assume that \(F_0 = \Omega \) .
Then \(F_2\)  satisfies
\( a(f) F_2 = - a^*(T(g)f) \Omega = - T(g)f, \quad f \in \mathcal {H}, \) 
i.e., \(F_2 = - \widehat{T(g)}\)  (Lemma REF ).
This observation implies Shale's
result that only for \(g \in \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) , i.e.,
\(\Vert g_2 \Vert _2 < \infty \) , equation
() has a solution \(\pi _s(g)\in \mathop {\rm U{}}\nolimits (S(\mathcal {H}))\) .
Step 3: Combining Lemma REF (i) with
Remark REF , we conclude that the exponential series
\(e^{-\widehat{T(g)}}\)  converges in \(S(\mathcal {H})\) 
for \(\Vert T(g)\Vert _2 < 1\) , which holds on an open \(\mathbf {1}\) -neighborhood in
\(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) .
Since the operators \(a(f)\)  act as derivations on \(S(\mathcal {H})_0\) , it follows that
\( a(f) e^{-\widehat{T(g)}}= - a(f) \widehat{T(g)} \vee e^{-\widehat{T(g)}}= - T(g)f \vee e^{- \widehat{T(g)}}= - a^*(T(g)f) e^{-\widehat{T(g)}}, \) 
so that \(e^{-\widehat{T(g)}}\)  satisfies (REF ).
We conclude that
\(\pi _s(g) \Omega = c(g) e^{-\widehat{T(g)}}, \quad c(g) \in \times .\) 
Choosing the operators \(\pi _s(g)\) , \(g \in \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) ,
in such a way that
\(c(g) = \langle \Omega _g, \Omega \rangle = \langle \pi _s(g)\Omega , \Omega \rangle > 0,\) 
it follows that
\(c(g) = \Vert e^{-\widehat{T(g)}}\Vert ^{-1}\) 
(cf. [1]}).
Step 4: Since the map
\(g \mapsto T(g) = g_2 g_1^{-1}\) 
is smooth in an identity neighborhood and the map
\( \Big \lbrace  A \in {\mathfrak {p}}_2 \colon  \Vert A\Vert _2 < 1\Big \rbrace  \rightarrow S(\mathcal {H}),\quad A \mapsto e^{\widehat{A}} \) 
is analytic (cf. Remark REF  and
the proof of Lemma REF (v) for \({\mathfrak {p}}_2\) ),
hence in particular smooth, it follows that
\(e^{-\widehat{T(g)}}\)  and hence also \(c(g)\)  are smooth in an identity neighborhood
of \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) .
From \(\langle \pi _s(g)^*\Omega , \Omega \rangle = \langle \Omega , \pi _s(g)\Omega \rangle > 0\) 
we further obtain \(\pi _s(g)^* = \pi _s(g^{-1})\) .
This implies that the map
\( (g,h) \mapsto \langle \pi _s(g)\pi _s(h)\Omega ,\Omega \rangle = \langle \pi _s(h)\Omega ,\pi _s(g)^*\Omega \rangle = \langle \pi _s(h)\Omega ,\pi _s(g^{-1})\Omega \rangle \) 
is smooth in an identity neighborhood. Now
Theorem REF  implies that
\(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  is a Lie group and that
\(\Omega \)  is a smooth vector for the corresponding unitary representation,
also denoted \(\pi _s\) .
Since \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\)  acts smoothly on \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\) , the
space \(S(\mathcal {H})^\infty \)  of smooth vectors is invariant under
\(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\) , so that the irreducibility of the Fock
representation of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  on \(S(\mathcal {H})\)  ([1]})
implies the smoothness of \(\pi _s\) .
Step 5: With Remark REF  we can now calculate a
suitable cocycle \(\eta \)  with \(F(g) := \pi _s(g)\Omega =c(g) e^{-\widehat{T(g)}}\)  by
\( \eta (x,y)= 2\mathop {{\rm Im}}\nolimits \langle {\tt d}F(\mathbf {1})x, {\tt d}F(\mathbf {1})y\rangle + i \langle {\tt d}F(\mathbf {1})[x,y],\Omega \rangle .\) 
As \(F(g)\)  only depends on \(T(g) = g_2 g_1^{-1}\) , we have
\(F(gu) = F(g)\)  for \(u \in \mathop {\rm U{}}\nolimits (\mathcal {H})\) , which leads to
\({\tt d}F(\mathbf {1})x = 0\)  for \(x \in {\mathfrak {u}}(\mathcal {H})\) . Hence
\(\eta (x,\cdot ) = 0\)  for \(x \in {\mathfrak {u}}(\mathcal {H})\) .
For \(x \in {\mathfrak {p}}_2\)  we find with
\(T(\exp x) = \cosh (x)\sinh (x)^{-1} = \tanh (x)\) 
the relation \({\tt d}T(\mathbf {1})x = x\) , and hence
\( {\tt d}F(\mathbf {1})x = {\tt d}c(\mathbf {1})(x) \Omega -\widehat{x}. \) 
Since \(c\)  is real-valued, this leads with Lemma REF (ii)
for \(x, y \in {\mathfrak {p}}_2\)  to
\(\eta (x,y)&= 2\mathop {{\rm Im}}\nolimits \langle {\tt d}c(\mathbf {1})(x) \Omega - \widehat{x},{\tt d}c(\mathbf {1})(y) \Omega - \widehat{y} \rangle = 2 \mathop {{\rm Im}}\nolimits \langle \widehat{x}, \widehat{y} \rangle \\&= \mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits (xy) = {\textstyle {\frac{1}{2}}}\mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits ([x,y]) = \frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x,y])\) 

because the trace of the symmetric operator \(xy+yx\)  is real
and \(\mathop {{\rm tr}}\nolimits ([x,y]) \in i {\mathbb {R}}\) .
\(\Box \)
The content of the preceding theorem is essentially known
(cf. [1]}, [4]}, [5]}),
although all references known to the author only discuss
the metaplectic representation
as a representation of a topological group
and not as a Lie group. In [1]}
and [7]} one finds quite explicit formulas for group
cocycles describing \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  as a central extension.
Remark 9.4  
The metaplectic representation of
\(G := \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  is not irreducible.
Since the representation of \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  on each \(S^n(\mathcal {H})\) 
is irreducible (Example REF ), every
\(G\) -invariant subspace is the direct sum of some \(S^n(\mathcal {H})\) ,
\(n \in {\mathbb {N}}_0\) .
The preceding proof immediately shows that
\(\pi (G)\Omega \subseteq S^{\rm even}(\mathcal {H}) := \widehat{\oplus }_{n \in {\mathbb {N}}_0} S^{2n}(\mathcal {H})\) 
and that all projections onto the subspaces \(S^{2n}(\mathcal {H})\) 
are non-zero.
Using the fact that the operators in
\({\tt d}\pi _s({\mathfrak {g}})\)  contain all multiplication operators
with elements \(\widehat{A}\) , \(A \in {\mathfrak {p}}_2\)  (cf. Lemma REF ),
and their adjoints ([1]}),
it easily follows that the representations of \(G\)  on the two
subspaces
\(S^{\rm even}(\mathcal {H})\)  and \(S^{\rm odd}(\mathcal {H})\)  are irreducible.

The metaplectic group
In this subsection we describe a convenient description
of the metaplectic group \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  as a Banach–Lie group.
For further details we refer to [9]}.
First we recall from [9]}
that
\(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})&:= \Big \lbrace  g = \begin{pmatrix}  a & b \\ c & d \end{pmatrix} \in \mathop {{\rm Sp}}\nolimits (\mathcal {H}) \subseteq \mathop {\rm U{}}\nolimits (\mathcal {H},\mathcal {H}) \colon  \\&\qquad \qquad \qquad \Vert b\Vert _2, \Vert c\Vert _2 < \infty ,\Vert a - \mathbf {1}\Vert _1, \Vert d - \mathbf {1}\Vert _1 < \infty \Big \rbrace \) 

carries the structure of a Banach–Lie group
with polar decomposition
\( \mathop {\rm U{}}\nolimits _1(\mathcal {H}) \times {\mathfrak {p}}_2 \rightarrow \mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H}), \quad {\mathfrak {p}}_2 = \Big \lbrace  \begin{pmatrix}  0 & b \\ b^* & 0 \end{pmatrix} \colon  b = b^\top \in B_2(\mathcal {H})\Big \rbrace \) 
(cf. [9]}).
The full unitary group \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  acts smoothly by conjugation
on \(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\) , so that we can form the semidirect product
\(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H})\) , and the multiplication map
\( \mu _1 \colon  \mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}),\quad (g,u) \mapsto gu \) 
is a quotient morphism of Banach–Lie groups with kernel
\( N := \lbrace  (g,g^{-1}) \colon  g \in \mathop {\rm U{}}\nolimits _1(\mathcal {H}) \rbrace , \) 
so that
\( \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) \cong (\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/N \) 
(cf. [9]}).
From the polar decomposition of
\(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\)  we derive that
\(\pi _1(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})) \cong \pi _1(\mathop {\rm U{}}\nolimits _1(\mathcal {H})) \cong {\mathbb {Z}}\) , hence the
existence of a unique 2-fold covering group
\(q \colon  \mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H})\rightarrow \mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\) . On the inverse
image \(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H})\)  of \(\mathop {\rm U{}}\nolimits _1(\mathcal {H})\)  in \(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\)  we then
have a unique character
\(\sqrt{\det } \colon  \widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \rightarrow \mbox{ with} \quad \mathop {\bf L{}}\nolimits (\sqrt{\det }) = \frac{1}{2i} \mathop {{\rm tr}}\nolimits .\) 
Next we observe that the smooth action of \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  on
\(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\) 
lifts to a smooth action on \(\mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H})\) . We also note that,
for \(\mathop {{\rm SU}}\nolimits _1(\mathcal {H}) := \ker (\det )\) , we have
\(\mathop {\rm U{}}\nolimits _1(\mathcal {H}) \cong \mathop {{\rm SU}}\nolimits _1(\mathcal {H}) \rtimes , where the determinantis the projection onto the second factor. Accordingly,\) U1(H) SU1(H)  with
\(q(g,t) = (g,t^2)\)  and \(\sqrt{\det }(g,t) = t\) .
Writing \(x = x_1 + x_2\)  for the decomposition of \(x \in \mathfrak {sp}(\mathcal {H})\) 
into linear and antilinear component,
we recall from Theorem REF  that
\( \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H}) \cong {\mathbb {R}}\oplus _\eta \mathfrak {sp}_{\rm res}(\mathcal {H})\quad \mbox{ with } \quad \eta (x,y) = \frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x_2, y_2]). \) 
This implies that
\( \sigma \colon  \mathfrak {sp}_{1,2}(\mathcal {H}) \rightarrow \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\cong {\mathbb {R}}\oplus _\eta \mathfrak {sp}_{\rm res}(\mathcal {H}), \quad \sigma (x) := \Big (\frac{1}{2i} \mathop {{\rm tr}}\nolimits (x_1), x\Big ) \) 
is a homomorphism of Banach–Lie algebras. Here we use that
\( [x,y]_1 = [x_1, y_1] + [x_2, y_2]\quad \mbox{ and } \quad \mathop {{\rm tr}}\nolimits ([x_1, y_1])=0. \) 
On the subgroup \(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \subseteq \mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H})\) ,
\(\sigma \)  integrates to the group homomorphism
\(\sigma _G \colon  \widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \rightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\subseteq \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) \times \mathop {\rm U{}}\nolimits (S(\mathcal {H})), \quad \sigma (g) := (q(g), \sqrt{\det }(g)\pi _s(g)), \) 
so that the
polar decomposition of \(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\)  implies that
\(\sigma \)  integrates to a morphism of Banach–Lie groups
\(\sigma _G \colon  \mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H}) \rightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}).\) 
Combining this map with the canonical inclusion
\(\mathop {\rm U{}}\nolimits (\mathcal {H}) \hookrightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) , the equivariance of
\(\sigma _G\)  under conjugation with unitary operators implies the
existence of a homomorphism
\( \mu \colon  \mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}),\quad (g,u) \mapsto \sigma _G(g)u.\) 
Proposition 9.5   The homomorphism \(\mu \) 
factors through an isomorphism
\( \overline{\mu }\colon  (\mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/\ker \mu \rightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) 
of connected Banach–Lie groups with
\(\ker \mu \cong \mathop {{\rm SU}}\nolimits _1(\mathcal {H}).\)

Proof.   
If \(\mu (g,u) = \mathbf {1}\) , then \(\sigma _G(g) = u^{-1}\)  implies that
\(g \in \widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H})\)  with \(q(g) = u^{-1}\) .
Now
\( \Omega = \pi _s(\mu (g,u)) \Omega = \sqrt{\det }(g)\Omega \) 
implies \(\sqrt{\det }(g) = 1\) . This shows that
\(\ker \mu = \lbrace (g,q(g)^{-1}) \colon  g \in \mathop {{\rm SU}}\nolimits _1(\mathcal {H})\rbrace ,\) 
and the assertion follows.
\(\Box \)
The inverse image of
the center \(Z \cong  of \) Spres(H)\( is\)\(\widehat{N} := \lbrace  (g,q(g)^{-1}) \colon  g \in \widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H})\rbrace .\)\(The inverse image \) U(H)\( of the subgroup \) U(H)\( in\) Spres(H)\( is of the form\begin{equation}\widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H}) \cong (\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/\ker \mu \cong Z \times \mathop {\rm U{}}\nolimits (\mathcal {H}).\end{equation}In particular, it splits as a central extension of Lie groups.\)
Semiboundedness of the metaplectic representation
In this subsection we show that the metaplectic representation
\((\pi _s, S(\mathcal {H}))\)  of \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  is semibounded and
determine the open invariant cone \(W_{\pi _s}\)  as the inverse
image of the canonical cone \(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\)  in
\(\mathfrak {sp}_{\rm res}(\mathcal {H})\) .
We start with a closer look at adjoint orbits of the
symplectic Lie algebra.
Lemma 9.6   If \((V,\omega )\)  is a finite dimensional
symplectic space and \(X \in \mathfrak {sp}(V,\omega )\)  is such that
the Hamiltonian function \(H_X(v) = {\textstyle {\frac{1}{2}}}\omega (Xv,v)\)  is indefinite, then
\(\mathop {{\rm conv}}\nolimits (\mathcal {O}_X) = \mathfrak {sp}(V,\omega )\) .

Proof.   
Let \({\mathfrak {g}}:= \mathfrak {sp}(V,\omega )\) . Then Proposition REF ,
combined with the uniqueness of the open invariant cone in
\(\mathfrak {sp}(V,\omega )\)  up to sign
(cf. [13]}),
implies that the set of semi-equicontinuous coadjoint
orbits coincides with the double cone
\(W_{\mathfrak {sp}(V,\omega )}^\star \cup - W_{\mathfrak {sp}(V,\omega )}^\star .\)
Using the Cartan–Killing form to identify \({\mathfrak {g}}\)  with its dual,
we accordingly see that any semi-equicontinuous
adjoint orbit is contained in the double cone
\( \overline{W_{\mathfrak {sp}(V,\omega )}} \cup - \overline{W_{\mathfrak {sp}(V,\omega )}}, \) 
i.e., the corresponding Hamiltonian function is either
positive or negative.
Finally, we observe that if \(x \in {\mathfrak {g}}\)  satisfies
\(\mathop {{\rm conv}}\nolimits (\mathcal {O}_x) \ne {\mathfrak {g}}\) , then \(B(\mathcal {O}_x)\)  is a non-zero invariant
cone and Proposition REF (i)
implies that it has interior points.
Therefore \(\mathcal {O}_x\)  is semi-equicontinuous because every finite dimensional
space is barrelled.
\(\Box \)
Lemma 9.7   The subspace
\({\mathfrak {p}}:= \lbrace  X \in \mathfrak {sp}(\mathcal {H}) \colon  IX = - XI\rbrace \) 
contains no non-trivial Lie algebra ideals, and the same holds
for the subspace
\({\mathfrak {p}}_2 := \lbrace  X \in \mathfrak {sp}_{\rm res}(\mathcal {H}) \colon  IX = - XI\rbrace = {\mathfrak {p}}\cap \mathfrak {sp}_{\rm res}(\mathcal {H}).\) 


Proof.   
First we show that \(X \in {\mathfrak {p}}\)  and \([X,[I,X]] = 0\)  implies
\(X = 0\) . Realizing \(\mathfrak {sp}(\mathcal {H})\)  as a closed subalgebra of
\(\mathop {{\mathfrak {gl} }}\nolimits (\mathcal {H}_ \cong \mathop {{\mathfrak {gl} }}\nolimits (\mathcal {H}\oplus \mathcal {H})\)  (cf. Example REF (h)),
we have
\( {\mathfrak {p}}= \Big \lbrace  \begin{pmatrix}  0 & a \\ a^* & 0 \end{pmatrix} \colon  a^\top = a \in B(\mathcal {H})\Big \rbrace \quad \mbox{ and } \quad I = \begin{pmatrix}  i & 0 \\ 0 & -i \end{pmatrix}. \) 
For \(X = \begin{pmatrix}  0 & a \\ a^* & 0 \end{pmatrix}\)  this leads to
\([X,[I,X]]&= \Big [\begin{pmatrix}  0 & a \\ a^* & 0 \end{pmatrix}, \begin{pmatrix}  0 & 2ia \\ -2ia^* & 0 \end{pmatrix}\Big ]= -4i\begin{pmatrix}  aa^* & 0 \\ 0& a^* a \end{pmatrix}.\) 

If this operator vanishes, then \(a^* a = 0\)  implies that
\(a = 0\) , so that \(X = 0\) .
If \({\mathfrak {i}}\subseteq {\mathfrak {p}}\)  is a Lie algebra ideal,
then we have for each \(X \in {\mathfrak {i}}\)  the relation
\([X,[I,X]] \in {\mathfrak {i}}\cap {\mathfrak {u}}(\mathcal {H}) = \lbrace 0\rbrace \) , so that
the preceding argument shows that \(X = 0\) . The same argument
applies to \({\mathfrak {p}}_2 = {\mathfrak {p}}\cap \mathfrak {sp}_{\rm res}(\mathcal {H})\) .
\(\Box \)
Lemma 9.8   Let \(X \in \mathfrak {sp}(\mathcal {H})\) . Then
the projection
\(p_{\mathfrak {k}}\colon  \mathfrak {sp}(\mathcal {H}) \rightarrow {\mathfrak {u}}(\mathcal {H})\) 
onto the \(-linear component satisfies\)\( p_{\mathfrak {k}}(\mathcal {O}_X) \subseteq \lbrace  y \in {\mathfrak {u}}(\mathcal {H}) \colon  i y \le 0 \rbrace = \overline{C_{{\mathfrak {u}}(\mathcal {H})}}, \)\(if and only if \) HX 0\(. A correspondingstatement holds for \) spres(H)\(.\)
Proof.   
Since \(p_{{\mathfrak {k}}}\)  is the fixed point projection for the
action of the torus \(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits I}\) , it preserves the closed convex cones
\(\pm \lbrace  Z \in \mathfrak {sp}(\mathcal {H}) \colon  H_Z \ge 0\rbrace \)  (Proposition REF ),
so that the relation
\(H_X \ge 0\)  implies \(i p_{\mathfrak {k}}(\mathcal {O}_X) \le 0\) 
(Theorem REF (i)).
Suppose, conversely, that \(i p_{\mathfrak {k}}(\mathcal {O}_X) \le 0\) .
If \(H_X \le 0\) , then the preceding argument
shows that \(i p_{\mathfrak {k}}(\mathcal {O}_X) \ge 0\) , which leads to
\(p_{\mathfrak {k}}(\mathcal {O}_X) = \lbrace 0\rbrace \) , so that the closed span of \(\mathcal {O}_X\)  is
an ideal of \({\mathfrak {g}}\)  contained in \({\mathfrak {p}}= \ker p_{\mathfrak {k}}\) .
In view of Lemma REF , this leads to \(X = 0\) .
We may therefore assume that \(H_X\)  is indefinite.
Hence there exists a finite dimensional
complex subspace \(\mathcal {H}_1 \subseteq \mathcal {H}\)  on which \(H_X\)  is indefinite.
Then we have a Hilbert space direct sum
\(\mathcal {H}= \mathcal {H}_1 \oplus \mathcal {H}_1^\bot \)  and accordingly we write
operators on \(\mathcal {H}\)  as \((2 \times 2)\) -block matrices:
\( X = \begin{pmatrix}  X_{11} & X_{12}\\ X_{21} & X_{22} \end{pmatrix}. \) 
For \(v \in \mathcal {H}_1\)  we then have
\(\mathop {{\rm Im}}\nolimits \langle Xv,v \rangle = \mathop {{\rm Im}}\nolimits \langle X_{11}v,v \rangle ,\) 
so that \(X_{11} \in \mathfrak {sp}(\mathcal {H}_1)\)  and \(H_{X_{11}}\)  is indefinite.
Identifying \(\mathop {{\rm Sp}}\nolimits (\mathcal {H}_1)\)  in the natural way with a subgroup
of \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\) , Lemma REF  implies that
\(\mathop {{\rm conv}}\nolimits (\mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits (\mathcal {H}_1)X))\)  contains an element
\(Y\)  with \(Y_{11} = - i \mathbf {1}\) .
From
\( p_{\mathfrak {k}}(Y)= \begin{pmatrix}  p_{\mathfrak {k}}(Y_{11}) & p_{\mathfrak {k}}(Y_{12})\\p_{\mathfrak {k}}(Y_{21}) & p_{\mathfrak {k}}(Y_{22}) \end{pmatrix}= \begin{pmatrix}  Y_{11} & p_{\mathfrak {k}}(Y_{12})\\p_{\mathfrak {k}}(Y_{21}) & p_{\mathfrak {k}}(Y_{22}) \end{pmatrix} \) 
we now derive that \(i p_{\mathfrak {k}}(Y) \lnot \le 0\) , contradicting
our assumption on \(X\) . This completes the proof of the
first assertion.
Since \(\mathop {{\rm Sp}}\nolimits (\mathcal {H}_1) \subseteq \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) , the preceding
argument also implies the assertion for the restricted
Lie algebra \(\mathfrak {sp}_{\rm res}(\mathcal {H})\) .
\(\Box \)
Now we are ready to show that the metaplectic representation
is semibounded. We write
\(q \colon  \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H}) \rightarrow \mathfrak {sp}_{\rm res}(\mathcal {H})\) 
for the quotient map and \({\mathfrak {z}}\)  for its kernel.
Since the central extension is trivial over the subalgebra
\({\mathfrak {u}}(\mathcal {H})\)  of \(\mathfrak {sp}_{\rm res}(\mathcal {H})\) , we have
\( \widehat{{\mathfrak {u}}}(\mathcal {H}) := q^{-1}({\mathfrak {u}}(\mathcal {H})) \cong {\mathfrak {z}}\oplus {\mathfrak {u}}(\mathcal {H}), \) 
where the \({\mathfrak {u}}(\mathcal {H})\) -complement is uniquely determined by the
property that it is the commutator algebra.
Here we use that the Lie algebra \({\mathfrak {u}}(\mathcal {H})\)  is perfect
(cf. [9]}).
Accordingly, we may identify \({\mathfrak {u}}(\mathcal {H})\)  in a natural
way with a subalgebra of \(\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\) .
Remark 9.9   The momentum set \(I_{\pi _s}\) 
is completely
determined by the restriction of its support function
\(s_{\pi _s}\)  to \(W_{\pi _s} = B(I_{\pi _s})^0\)  (Remark REF (b))
which is invariant under the adjoint action.
We shall see below that \(q(W_{\pi _s}) \subseteq W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\) ,
so that Theorem REF (ii) leads to
\( W_{\pi _s} \subseteq \mathop {{\rm Ad}}\nolimits (\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}))({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}), \) 
where \({\mathfrak {z}}\cong {\mathbb {R}}\)  denotes the center. This
in turn entails that \(I_{\pi _s}\)  is already determined
by the restriction of \(s_{\pi _s}\)  to \({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) ,
which we determine below.
This restriction is the support
function for the momentum set of the restriction of
\({\pi _s}\)  to the subgroup \(\widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H}) \cong \mathop {\rm U{}}\nolimits (\mathcal {H})\) 
(Proposition REF (iii)), which is a
direct sum of the representations on the subspaces \(S^n(\mathcal {H})\) , on
which we have
\( {\pi _s}(z,g)(v_1 \vee \cdots \vee v_n)= z gv_1 \vee \cdots \vee g v_n \quad \mbox{ for } \quad v_1,\ldots , v_n \in \mathcal {H}. \) 
For the representation of \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  on \(S^n(\mathcal {H})\) , the momentum
set \(I_{S^n(\mathcal {H})}^{\mathop {\rm U{}}\nolimits (\mathcal {H})}\)  is simply given by
\( I_{S^n(\mathcal {H})}^{\mathop {\rm U{}}\nolimits (\mathcal {H})} = n I_{\mathcal {H}}^{\mathop {\rm U{}}\nolimits (\mathcal {H})} \) 
(Example REF (b)), so that
\( I_{\pi _s}^{\widehat{U}(\mathcal {H})}= \overline{\mathop {{\rm conv}}\nolimits }\big (\bigcup _{n \in {\mathbb {N}}_0} \lbrace 1\rbrace  \times n I_\mathcal {H}^{\mathop {\rm U{}}\nolimits (\mathcal {H})}\big ), \) 
which is a convex cone with vertex \((1,0)\) , and for
\(x \in {\mathfrak {u}}(\mathcal {H})\) , we have
\(s_{\pi _s}(t,x) ={\left\lbrace \begin{array}{ll}-t & \text{ for } ix \le 0 \\\infty & \text{ else}. \\\end{array}\right.}\) 
The representation of \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  on
\(S(\mathcal {H})\)  decomposes into two irreducible pieces
\( S^{\rm even}(\mathcal {H}) = \widehat{\oplus }_{n \in {\mathbb {N}}_0} S^{2n}(\mathcal {H})\quad \mbox{ and } \quad S^{\rm odd}(\mathcal {H}) = \widehat{\oplus }_{n \in {\mathbb {N}}_0} S^{2n+1}(\mathcal {H}) \) 
(Remark REF ).
Writing \(\pi _\pm \)  for the corresponding representations, we obtain
\( I_{\pi _+}^{\widehat{U}(\mathcal {H})} = I_{{\pi _s}}^{\widehat{U}(\mathcal {H})}= \overline{\mathop {{\rm conv}}\nolimits }\Big (\bigcup _{n \in {\mathbb {N}}_0} \lbrace 1\rbrace  \times 2n I_\mathcal {H}^{\mathop {\rm U{}}\nolimits (\mathcal {H})}\Big ), \) 
and
\( I_{\pi _-}^{\widehat{U}(\mathcal {H})}= \overline{\mathop {{\rm conv}}\nolimits }\Big (\bigcup _{n \in {\mathbb {N}}_0} \lbrace 1\rbrace  \times (2n+1)I_\mathcal {H}^{\mathop {\rm U{}}\nolimits (\mathcal {H})}\Big ). \) 
Now it is easy to derive that
\(W_{\pi _\pm } = W_{\pi _s}\) , \(s_{\pi _+} = s_{\pi _s}\) , and that
\( s_{\pi _-}(t,x) = -t + \sup ({\rm Spec}(ix)) \quad \mbox{ for } \quad ix \le 0. \) 

Theorem 9.10  
The metaplectic representation
\(({\pi _s}, S(\mathcal {H}))\)  of
\(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}))\)  is semibounded with
\( W_{\pi _s} = W_{\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})} := q^{-1}(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}). \) 


Proof.   
From Example REF  we know that
\( s_{\pi _s}(x) \le 0 \quad \mbox{ for } \quad x \in C_{{\mathfrak {u}}(\mathcal {H})},\quad \mbox{ so that } \quad s_{\pi _s}(t,x) = -t + s_{\pi _s}(x) < \infty \) 
for \((t,x) \in {\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) .
From the invariance of \(s_{\pi _s}\)  under the adjoint action
and Theorem REF (ii) it now follows that
\(\mathop {{\rm Ad}}\nolimits (\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}))({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})})= q^{-1}(\mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}))C_{{\mathfrak {u}}(\mathcal {H})})= q^{-1}(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}) \subseteq B(I_{\pi _s})\) 
(cf. Theorem REF (ii)).
In particular, \(B(I_{\pi _s})\)  has interior points,
so that \({\pi _s}\)  is semibounded with \(W_{\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})} \subseteq W_{\pi _s}\) .
To prove equality, we note that (REF ) in
Remark REF  implies that
\( W_{\pi _s} \cap {\mathfrak {u}}(\mathcal {H}) \subseteq \lbrace  x \in {\mathfrak {u}}(\mathcal {H}) \colon  s_{\pi _s}(x) < \infty \rbrace = \lbrace  x \in {\mathfrak {u}}(\mathcal {H}) \colon  ix \le 0\rbrace  = \overline{C_{{\mathfrak {u}}(\mathcal {H})}}. \) 
As \(W_{\pi _s}\)  is invariant under the projection
\(p_{\mathfrak {k}}\colon  \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H}) \rightarrow \widehat{{\mathfrak {u}}}(\mathcal {H})\) 
onto the fixed point space of the compact group
\(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits I}\)  (Proposition REF ),
we obtain for each \(x \in W_{\pi _s}\)  the relation
\(i p_{\mathfrak {k}}(\mathcal {O}_{q(x)})\le 0,\) 
so that Lemma REF  leads to
\(q(x) \in \overline{W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}}\) . As \(W_{\pi _s}\)  is open and
\(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\)  coincides with the interior of its closure
(Lemma REF ), it follows that
\( q(W_{\pi _s}) \subseteq W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}. \) 
Combining this with (REF ),
we obtain \(q(W_{\pi _s}) = W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}.\)
As the center \({\mathfrak {z}}\cong {\mathbb {R}}\)  of \(\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\) 
acts by multiples of the identity on \(S(\mathcal {H})\) ,
we have \({\mathfrak {z}}\subseteq H(W_{\pi _s})\) , which finally leads to
\(W_{\pi _s} = q^{-1}(q(W_{\pi _s})) = W_{\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})}.\) 
\(\Box \)
The momentum set of the metaplectic representation
Now that we have determined the cone \(W_{\pi _s}\)  for the metaplectic
representation, we now apply the tools from Section 
to determine the momentum set for the representation of the
central extension \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  acting on
\(S(\mathcal {H})\) . In particular, we show that the momentum set is the
closed convex hull of a single coadjoint orbit.
Definition 9.11  We define
\(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\) 

as the quotient of the semidirect product
\( \mathop {{\rm Heis}}\nolimits (\mathcal {H}) \rtimes _\alpha \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}), \quad \alpha (g,u)((t,v)) := (t,gv), \) 
by the central subgroup
\( S := \lbrace  (t,(\mathbf {1},z\mathbf {1})) \in {\mathbb {R}}\times Z \colon  e^{it}z = 1\rbrace  \cong {\mathbb {R}}\) 
which acts trivially on \(S(\mathcal {H})\) .
This means that \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is a central extension
of \(\mathcal {H}\rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\)  by \(. The correspondingLie algebra cocycle is given by\)\(((v,x), (v^{\prime },x^{\prime })) \mapsto \mathop {{\rm Im}}\nolimits \langle v,v^{\prime }\rangle + \eta (x,x^{\prime })= \mathop {{\rm Im}}\nolimits \langle v,v^{\prime }\rangle + {\textstyle {\frac{1}{2}}}\mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits (x_2 x^{\prime }_2) \)\((cf.\ Theorem~\ref {thm:metasmooth}).\)
Remark 9.12 
The representations \(W\)  of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  and \(\pi _s\)  of
\(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  combine to a representation, also denoted
\(\pi _s\)  of
\(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  on \(S(\mathcal {H})\) . Since \((W,S(\mathcal {H}))\)  is irreducible,
the extension to \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is also irreducible.
To see that it is smooth, it suffices to show that the function
\( \mathop {{\rm Heis}}\nolimits (\mathcal {H}) \rtimes \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\rightarrow  \quad (h,g) \mapsto \langle W(h) \pi _s(g)\Omega , \Omega \rangle = \langle \pi _s(g)\Omega , W(h)^{-1}\Omega \rangle \) 
is smooth (Theorem REF ).
This follows from the smoothness
of the vector \(\Omega \)  for \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  and \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) 
(cf. (REF ) and Theorem REF ).

Proposition 9.13  The metaplectic representation \((\pi _s, S(\mathcal {H}))\) 
of \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is semibounded with
\( W_{\pi _s} = q^{-1}(\mathcal {H}\times W_{\mathfrak {sp}_{\rm res}(\mathcal {H})})= \mathop {{\rm Ad}}\nolimits (\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H}))({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}), \) 
where \(q \colon  \widehat{\mathfrak {hsp}}_{\rm res}(\mathcal {H}) \rightarrow \mathcal {H}\rtimes \mathfrak {sp}_{\rm res}(\mathcal {H})\) 
is the quotient map whose kernel is the center \({\mathfrak {z}}\cong {\mathbb {R}}\) .

Proof.   
Let
\(p \colon  \mathfrak {hsp}_{\rm res}(\mathcal {H}) = \mathfrak {heis}(\mathcal {H}) \rtimes \mathfrak {sp}_{\rm res}(\mathcal {H})\rightarrow \mathcal {H}\rtimes \mathfrak {sp}_{\rm res}(\mathcal {H}), \quad (t,v,x) \mapsto (v,x) \) 
denote the quotient map, so that Theorem REF (iv) implies that
\( W_{\mathfrak {hsp}_{\rm res}(\mathcal {H})} = p^{-1}(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})})= \mathop {{\rm Ad}}\nolimits (\mathop {\rm HSp{}}\nolimits _{\rm res}(\mathcal {H})) W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}. \) 
Modulo the center, this relation leads to
\(\mathcal {H}\rtimes W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}= \mathop {{\rm Ad}}\nolimits (\mathcal {H}\rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}))W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}= \mathop {{\rm Ad}}\nolimits (\mathcal {H}\rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}))C_{{\mathfrak {u}}(\mathcal {H})}.\) 
From Theorem REF  we know that
\(s_{\pi _s}(x)\)  is finite if \(x \in {\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) ,
so that (REF ) implies that
it is also finite if \(q(x) \in \mathcal {H}\rtimes W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\) , i.e.,
\( W_{\pi _s} \supseteq q^{-1}(\mathcal {H}\rtimes W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}). \) 
This proves already that \(\pi _s\)  is semibounded
because \(W_{\pi _s}\)  has interior points (Proposition REF ).
We further conclude that \(\mathfrak {heis}(\mathcal {H}) \subseteq H(W_{\pi _s})\) , so that
we obtain with Theorem REF  that
\(W_{\pi _s}&= \mathfrak {heis}(\mathcal {H}) + (W_{\pi _s} \cap \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H}))\subseteq \mathfrak {heis}(\mathcal {H}) + q^{-1}(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}) \\&= q^{-1}(\mathcal {H}\times W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}).\) 

This proves the desired equality.
Finally, we note that (REF ) implies that every element
\(x \in W_{\pi _s}\)  is conjugate to an element \(y \in q^{-1}(C_{{\mathfrak {u}}(\mathcal {H})})\) ,
which means that \(y \in {\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) .
\(\Box \)
Theorem 9.14  
The momentum set \(I_{\pi _s}\)  of the metaplectic representation
of \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is the closed convex hull of the coadjoint
orbit of \(\lambda := \Phi _{\pi _s}([\Omega ])\) , and this linear functional
is given by
\(\lambda (t,x) = t\quad \mbox{ on } \quad \widehat{\mathfrak {hsp}}_{\rm res}(\mathcal {H}) = {\mathbb {R}}\oplus (\mathcal {H}\rtimes \mathfrak {sp}_{\rm res}(\mathcal {H})).\) 


Proof.   
Since the metaplectic representation of
\(G := \widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is irreducible,
we want to apply Theorem REF .
For the subgroup \(K := \widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H})\subseteq G\)  with Lie algebra
\({\mathfrak {k}}= {\mathfrak {z}}\oplus {\mathfrak {u}}(\mathcal {H})\) , we have
\( {\mathfrak {g}}_{\mathfrak {p}}_+ \oplus {\mathfrak {k}}_{\mathfrak {p}}_-,\quad \mbox{ where } \quad {\mathfrak {p}}_\pm = \ker (\mathop {{\rm ad}}\nolimits I \mp i\mathbf {1})\oplus \ker (\mathop {{\rm ad}}\nolimits I \mp 2i\mathbf {1}), \) 
\(I \in {\mathfrak {u}}(\mathcal {H})\)  is the multiplication with \(i\)  on \(\mathcal {H}\) ,
\( \mathfrak {heis}(\mathcal {H})_ {\mathfrak {z}}_\mathcal {H}_+ \oplus \mathcal {H}_-, \quad \mathcal {H}_\pm := \ker (\mathop {{\rm ad}}\nolimits I \mp i\mathbf {1})\subseteq \mathcal {H}_ \) 
Since the Lie algebra \({\mathfrak {k}}\)  of \(K\)  is complemented by the closed
subspace \(\mathcal {H}\oplus {\mathfrak {p}}_2\)  in \({\mathfrak {g}}\) , the coset space
\(G/K\)  carries the structure of a Banach homogeneous space.
Moreover, the closed \(K\) -invariant subalgebra
\({\mathfrak {k}}_{\mathfrak {p}}_+\)  determines on \(G/K\)  the structure of a
complex manifold for which the tangent space in the base point
can be identified with \({\mathfrak {g}}_({\mathfrak {k}}_ {\mathfrak {p}}_+) \cong {\mathfrak {p}}_-\) 
([15]}).
Let \(\Omega \in S^0(\mathcal {H}) \subseteq S(\mathcal {H})\)  be the vacuum vector.
Then \(\Omega \)  is a smooth vector (Remark REF ),
which is an eigenvector for \(K\) . On \(S(\mathcal {H})\) , the operator
\(-i{\tt d}\pi (I)\)  is diagonal, and \(S^n(\mathcal {H})\)  is the eigenspace corresponding
to the eigenvalue \(n\) . As \(-i\mathop {{\rm ad}}\nolimits I\)  acts on \({\mathfrak {p}}_-\)  with the
eigenvalues \(-1\)  and \(-2\) , it follows that
\({\tt d}\pi _s({\mathfrak {p}}_-)\Omega = \lbrace 0\rbrace ,\) 
so that Theorem REF  provides a holomorphic equivariant map
\( \eta \colon  G/K\rightarrow {\mathbb {P}}(S(\mathcal {H})^{\prime }), \quad g \mapsto \langle \cdot , \pi _s(g)^{-1} \Omega \rangle \) 
and a realization of \((\pi _s, S(\mathcal {H}))\)  by holomorphic sections of a
complex line bundle over \(G/K\) . Since the representation
\((\pi _s, S(\mathcal {H}))\)  of \(G\)  is irreducible, \(\Omega \)  is a cyclic vector.
It remains to show that for each \(x \in W_\pi \)  the flow generated by
\(-x\)  on \(G/K\)  extends holomorphically to \(+\) . Since every such
element \(x\)  is conjugate to an element of \({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) , we may
w.l.o.g. assume that \(x \in C_{{\mathfrak {u}}(\mathcal {H})}\) .
To get more information on \(G/K\) ,
we note that the choice of the complex
structure implies that \(G/K\)  contains
\(\mathop {{\rm Heis}}\nolimits (\mathcal {H})/(K \cap \mathop {{\rm Heis}}\nolimits (\mathcal {H})) \cong \mathcal {H}_-\cong \mathcal {H}^{\prime }\) 
as a complex submanifold.
The translation action of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  by
\((t,v).z := v + z\)  on \(\mathcal {H}\)  factors through an action of the additive group
\(\mathcal {H}\)  and extends naturally to a holomorphic action of the
complexified group \(\mathcal {H}_\mathcal {H}_+ \times \mathcal {H}_-\) .
As the \(G\) -action on \(G/K\)  induces a transitive
action on the set of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\) -orbits on \(G/K\) ,
the action of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  extends to a holomorphic action of
\(\mathop {{\rm Heis}}\nolimits (\mathcal {H})_ on \) G/K\(. Therefore the action of the one-parameter group\) t (-tx)\( extends to a holomorphic action of \) +\(on \) G/K\( if and only if the same holds for the actionon the quotient space \) Spres(H)/U(H)\(.As we have seen in Example~\ref {ex:liegrp}(h),this space can be identified with the symmetric Hilbert domain\)\( \mathcal {D}_s = \lbrace  z \in B_2(\mathcal {H}) \colon  z^\top = z, zz^* <\!\!< 1 \rbrace , \)\(but here it is endowed with the opposite complex structure.\) The group \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  acts on \(\mathcal {D}_s\)  by
\(u.z = uzu^\top \) , and for \(x \in C_{{\mathfrak {u}}(\mathcal {H})}\) 
we obtain for our choice of complex structure the relation
\(e^{-ix}.z = e^{ix} ve^{ix^\top }.\) 
Now \(\Vert e^{ix}\Vert  = \Vert e^{ix^\top }\Vert < 1\) 
implies that the action of the one-parameter group
\(t \mapsto \exp (-tx)\)  extends to \(+\) , and this completes the
proof.
Finally, we derive from Theorem REF (b) that
\(I_{\pi _s} = \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_\lambda )\) 
holds for
\( \lambda (x) = - i \langle {\tt d}\pi _s(x)\Omega , \Omega \rangle . \) 
On \({\mathfrak {k}}= \widehat{{\mathfrak {u}}}(\mathcal {H})\)  we have \(\lambda (t,x) = t\) , and \(\lambda \)  vanishes
on \({\mathfrak {p}}_ because \) p-\( annihilates \)\(, and\)ds(p+), = , ds(p-)= 0\(.This completes the proof.\hfill \Box \)
With similar and even easier arguments as in the proof of the
preceding theorem, we also obtain:
Theorem 9.15  
The momentum set \(I_{\pi _s^+}\)  of the even metaplectic representation
\((\pi _s^+, S^{\rm even}(\mathcal {H}))\)  of \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) 
is the closed convex hull of the coadjoint
orbit of \(\Phi _{\pi _s^+}([\Omega ])\) .

The spin representation
In this section we take a closer look
at the spin representation
\((\pi _a, \Lambda (\mathcal {H}))\)  of the
central extension \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  of \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\) 
on the fermionic Fock space \(\Lambda (\mathcal {H})\) .
This representation arises from self-intertwining
operators of the Fock representation of the \(C^*\) -algebra
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) . Here we also show that \(\pi _a\)  is semibounded and
determine the corresponding cone \(W_{\pi _a}\) .
For the irreducible representation of the
identity component on the even part
\(\Lambda ^{\rm even}(\mathcal {H})\) , we show that the momentum set
is the weak-\(*\) -closed convex hull of a single coadjoint orbit.
Semiboundedness of the spin representation
Let \(\mathcal {H}\)  be a complex Hilbert space and
write \(\lbrace a,b\rbrace  := ab + ba\)  for the anticommutator of two elements
of an associative algebra. The
CAR-algebra of \(\mathcal {H}\)  is a \(C^*\) -algebra
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) , together with a continuous antilinear map
\(a \colon  \mathcal {H}\rightarrow \mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  satisfying the
canonical anticommutation relations
\(\lbrace a(f), a(g)^*\rbrace  = \langle g,f \rangle \mathbf {1}\quad \hbox{ and } \quad \lbrace a(f),a(g)\rbrace  = 0\quad \mbox{ for } \quad f, g \in \mathcal {H}\) 
and which is generated by the image of \(a\) . This determines
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  up to natural isomorphism ([16]}).
We also write \(a^*(f) := a(f)^*\) , which defines a complex
linear map \(a^* \colon  \mathcal {H}\rightarrow \mathop {{\rm CAR}}\nolimits (\mathcal {H})\) .
The orthogonal group \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  of the underlying real
Hilbert space
\((\mathcal {H}_{\mathbb {R}},\beta ), \quad \beta (v,w) = \mathop {{\rm Re}}\nolimits \langle v,w\rangle ,\) 
acts by automorphisms on this \(C^*\) -algebra as follows. Writing
a real linear isometry as
\(g = g_1 + g_2\) , where \(g_1\)  is linear and \(g_2\)  is antilinear, the
relations \(gg^* = g^*g = \mathbf {1}\)  turn into
\( g_1 g_1^* + g_2 g_2^* = \mathbf {1} = g_1^*g_1 + g_2^* g_2\quad \mbox{ and } \quad g_1 g_2^* + g_2 g_1^* = 0 = g_2^* g_1 + g_1^*g_2. \) 
These relations imply that
\( a_g \colon  \mathcal {H}\rightarrow \mathop {{\rm CAR}}\nolimits (\mathcal {H}), \quad f \mapsto a(g_1 f) + a^*(g_2 f) \) 
satisfies the same anticommutation relations, so that the
universal property of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  implies the existence of a unique
automorphism \(\alpha _g\)  with
\( \alpha _g(a(f)) = a_g(f) \quad \mbox{ for } \quad f \in \mathcal {H}.\) 
These automorphisms of the \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  are called
Bogoliubov automorphisms. They define an action of
\(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) . In particular, the unitary group
\(\mathop {\rm U{}}\nolimits (\mathcal {H}) \subseteq \mathop {\rm O{}}\nolimits (\mathcal {H})\)  acts on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  by
\(\alpha _g(a(f)) = a(gf)\)  for \(f \in \mathcal {H}\) .
Remark 10.1  
The \(C^*\) -algebra \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  has a natural irreducible
representation
\((\pi _0, \Lambda (\mathcal {H}))\) 
on the antisymmetric Fock space \(\Lambda (\mathcal {H})\)  ([16]}).
The image \(a_0(f) := \pi _0(a(f))\) 
acts by \(a_0(f)\Omega =0\)  and
\( a_0(f)(f_1 \wedge \cdots \wedge f_n)= \sum _{j = 1}^n (-1)^{j-1} \langle f_j, f \rangle f_1 \wedge \cdots \wedge f_{j-1} \wedge f_{j+1} \wedge \cdots \wedge f_n.\) 
Accordingly, we have
\( a_0^*(f)\Omega = f \quad \mbox{ and } \quad a_0^*(f)(f_1 \wedge \cdots \wedge f_n)= f \wedge f_1 \wedge \cdots \wedge f_n. \) 
Let \(Iv = iv\)  denote the complex structure on \(\mathcal {H}\) . The action of the
restricted orthogonal group
\(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\colon = \lbrace  g \in \mathop {\rm O{}}\nolimits (\mathcal {H}) \colon  \Vert [g,I]\Vert _2 < \infty \rbrace  \) 
on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  preserves the equivalence class of the representation
\(\pi _0\) ,
so that there is a projective unitary representation
\(\pi _a \colon  \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}) \rightarrow \mathop {\rm U{}}\nolimits (\Lambda (\mathcal {H}))\) , called the
spin representation, satisfying
\(\pi _a(g) \pi _0(A)\pi _a(g)^* = \pi _0(\alpha _g A) \quad \mbox{ for }\quad g \in \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}), A \in \mathop {{\rm CAR}}\nolimits (\mathcal {H})\) 
(cf. [1]}, [19]}).

In analogy with Theorem REF  we here obtain:
Theorem 10.2 
\(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  is a Lie group and
the spin representation is smooth.
A Lie algebra cocycle \(\eta \)  defining \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) 
as an extension of \({\mathfrak {o}}_{\rm res}(\mathcal {H})\)  by \({\mathbb {R}}\)  is given by
\( \eta (x,y) = -\frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x_2, y_2]). \) 


Proof.   
With completely analogous arguments as in the proof of
Theorem REF  we find
with Appendix  that any
\(g \in \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  for which \(g_1\)  is invertible
has a unique lift \(\pi _a(g) \in \mathop {\rm U{}}\nolimits (\Lambda (\mathcal {H}))\)  with
\( \pi _a(g) \Omega = c(g) e^{-\widehat{T}(g)} \quad \mbox{ for } \quad T(g) := g_2 g_1^{-1} \in \mathop {{\rm Aherm}}\nolimits (\mathcal {H})_a,c(g) = \Vert e^{-\widehat{T}(g)}\Vert ^{-1} \) 
(cf. Lemma REF ).
This implies that \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  is a Lie group,
its representation \(\pi _a\)  on \(\Lambda (\mathcal {H})\)  is smooth, and
that \(\Omega \)  is a smooth vector.
Since \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  acts smoothly on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) ,
the space \(\Lambda (\mathcal {H})^\infty \)  of smooth vectors is invariant under
\(\pi _0(\mathop {{\rm CAR}}\nolimits (\mathcal {H}))\) , so that the irreducibility of the
representation of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  on \(\Lambda (\mathcal {H})\) 
implies the smoothness of \(\pi _a\) .
For the Lie algebra cocycle defining \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) ,
we find as in the proof of Theorem REF 
with Lemma REF (iii)
\(\eta (x,y)= 2 \mathop {{\rm Im}}\nolimits \langle \widehat{x}_2, \widehat{y}_2 \rangle = -\mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits (x_2y_2) = -\frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x_2,y_2]).\) 

\(\Box \)
Remark 10.3   As in Subsection REF ,
we obtain a Banach–Lie group
\(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})&:= \Big \lbrace  g = g_1 + g_2 \in \mathop {\rm O{}}\nolimits (\mathcal {H})\colon \Vert g_2 \Vert _2 < \infty , \Vert \mathbf {1} - g_1\Vert _1 < \infty \rbrace \) 

with
\( \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}) \cong (\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/N,\quad N \cong \mathop {\rm U{}}\nolimits (\mathcal {H}) \cap \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H}) = \mathop {\rm U{}}\nolimits _1(\mathcal {H}) \) 
(cf. [9]}). From [9]}
we recall that \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  has two connected components
and that its fundamental group is trivial.
On the other hand
\( \pi _0(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \cong {\mathbb {Z}}/2{\mathbb {Z}}\cong \pi _1(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \) 
([22]}, [9]}),
so that there exists a simply connected 2-fold covering group
\(q \colon  {\rm Spin}_{1,2}(\mathcal {H})\rightarrow \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})_0\) 
(cf. [24]} for a discussion of the smaller group
\({\rm Spin}_1(\mathcal {H})\)  covering \(\mathop {{\rm SO}}\nolimits _1(\mathcal {H}) = \mathop {\rm O{}}\nolimits _1(\mathcal {H})_0\) ).
As the
inclusion \(\mathop {\rm U{}}\nolimits _{1}(\mathcal {H}) \rightarrow \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})\)  induces a surjective homomorphism
\( \pi _1(\mathop {\rm U{}}\nolimits _1(\mathcal {H})) \cong {\mathbb {Z}}\rightarrow \pi _1(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \cong {\mathbb {Z}}/2, \) 
\(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) := q^{-1}(\mathop {\rm U{}}\nolimits _1(\mathcal {H}))\)  is the unique
2-fold connected covering of
\(\mathop {\rm U{}}\nolimits _1(\mathcal {H})\)  from Subsection REF .
We further have an embedding
\(\sigma \colon  {\mathfrak {o}}_{1,2}(\mathcal {H}) \rightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H}), \quad \sigma (x) := \Big (-\frac{1}{2i} \mathop {{\rm tr}}\nolimits (x_1), x\Big )\) 
of Banach–Lie algebras.
On the subgroup \(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \subseteq {\rm Spin}_{1,2}(\mathcal {H})\) ,
\(\sigma \)  integrates to a group homomorphism
\(\sigma _G(g) := (g, \sqrt{\det }(g)^{-1}\pi _s(g))\) 
(cf. Subsection REF )
and \(\sigma \)  integrates to a morphism of Banach–Lie groups
\(\sigma _G \colon  {\rm Spin}_{1,2}(\mathcal {H}) \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H}).\) 
Combining this map with the canonical inclusion
\(\mathop {\rm U{}}\nolimits (\mathcal {H}) \hookrightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) , the equivariance of
\(\sigma _G\)  under conjugation with unitary operators implies the
existence of a homomorphism
\( \mu \colon  {\rm Spin}_{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H}),\quad (g,u) \mapsto \sigma _G(g)u.\) 

The following proposition is proved as Proposition REF ,
using the representation \(\pi _a\)  instead of \(\pi _s\) .
Proposition 10.4  The homomorphism \(\mu \)  factors through an
isomorphism
\( \overline{\mu }\colon  ({\rm Spin}_{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/\ker \mu \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\) 
of connected Banach–Lie groups with
\(\ker \mu \cong \mathop {{\rm SU}}\nolimits _1(\mathcal {H}).\)

Remark 10.5  
The embedding \({\mathfrak {o}}_{1,2}(\mathcal {H}) \hookrightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) 
restricts in particular to an embedding \({\mathfrak {o}}_1(\mathcal {H}) \hookrightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) . One can show that the operators
\({\tt d}\pi _a(x)\)  are bounded for \(x \in {\mathfrak {o}}_1(\mathcal {H})\) 
(cf. [25]}, [26]}), so that we obtain
a morphism of Banach–Lie algebras
\( {\mathfrak {o}}_1(\mathcal {H}) \rightarrow {\mathfrak {u}}(\Lambda (\mathcal {H})), \quad x \mapsto {\tt d}\pi _a(x) - \frac{1}{2} \mathop {{\rm tr}}\nolimits (x_1)\mathbf {1}. \) 

Definition 10.6  To determine the open cone \(W_{\pi _a}\) 
for the spin representation, we have to take a closer
look at natural cones in \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) .
Let \({\mathfrak {u}}_\infty (\mathcal {H})\)  denote the ideal of compact skew-hermitian
operators in \({\mathfrak {u}}(\mathcal {H})\)  and recall the Calkin algebra
\(\mathop {{\rm Cal}}\nolimits (\mathcal {H}) := B(\mathcal {H})/B_\infty (\mathcal {H})\) , where
\(B_\infty (\mathcal {H})\)  denotes the ideal of compact operators
in the \(C^*\) -algebra \(B(\mathcal {H})\) . Then the surjection
\(B(\mathcal {H}) \rightarrow \mathop {{\rm Cal}}\nolimits (B(\mathcal {H}))\)  induces an isomorphism
\( {\mathfrak {u}}(\mathcal {H})/{\mathfrak {u}}_\infty (\mathcal {H}) \cong {\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H})). \) 
Next we note that
\(\lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  x^* - x \in {\mathfrak {u}}_\infty (\mathcal {H})\rbrace \) 
is a closed ideal of \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) 
which defines a quotient morphism
\(q \colon  {\mathfrak {o}}_{\rm res}(\mathcal {H}) \rightarrow {\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H})).\) 
We thus obtain an open invariant cone in \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) 
by
\( W_{{\mathfrak {o}}_{\rm res}(\mathcal {H})} := q^{-1}(C_{{\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H}))})= \lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  -i q(x) <\!\!< 0 \rbrace  \) 
(cf. Example REF ). We write
\(W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})}\)  for its inverse image in the
central extension \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) .

Theorem 10.7  
The spin representation \((\pi _a, \Lambda (\mathcal {H}))\)  of
the connected Lie group \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  is semibounded
with \(W_{\pi _a} = W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})}\) .

Proof.   
First we note that under
\(\widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H})\cong \mathop {\rm U{}}\nolimits (\mathcal {H})\)  the spin representation decomposes
into the irreducible subspaces \(\Lambda ^n(\mathcal {H})\)  with
\( {\pi _a}(z,g)(v_1 \wedge \cdots \wedge v_n)= z gv_1 \wedge \cdots \wedge g v_n \quad \mbox{ for } \quad v_1,\ldots , v_n \in \mathcal {H}. \) 
If \(i x \le 0\) , then the corresponding operator
on \(\Lambda ^n(\mathcal {H}) \subseteq \mathcal {H}^{\otimes n}\)  is also
\(\le 0\) , and this implies that
\( s_{\pi _a}(t,x) = -t + s_{\pi _a}(x) \le - t < \infty \quad \mbox{ for } \quad x \in C_{{\mathfrak {u}}(\mathcal {H})}. \) 
Writing \({\mathfrak {z}}\cong {\mathbb {R}}\)  for the center of \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) ,
we thus find that
\({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})} \subseteq B(I_{\pi _a}).\)
Next we consider the decomposition of \({\mathfrak {g}}= \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})= {\mathfrak {k}}\oplus {\mathfrak {p}}_2\) , where \({\mathfrak {k}}= {\mathfrak {z}}\times {\mathfrak {u}}(\mathcal {H})\)  and
\({\mathfrak {p}}_2 = \lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  Ix = - xI\rbrace .\) 
For the map
\( F \colon  {\mathfrak {k}}\times {\mathfrak {p}}_2 \rightarrow {\mathfrak {g}}, \quad (x,y) \mapsto e^{\mathop {{\rm ad}}\nolimits y} x \quad \mbox{ we then have }\quad {\tt d}F(x,0)(a,b) = a + [b,x], \) 
which is invertible if and only if \(\mathop {{\rm ad}}\nolimits x \colon  {\mathfrak {p}}_2 \rightarrow {\mathfrak {p}}_2\)  is
invertible. This is in particular the case for
\(x = I\) , the complex structure of \(\mathcal {H}\) . Therefore the image of
\(({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}) \times {\mathfrak {p}}_2\)  has interior points,
and since it is contained in the invariant subset
\(B(I_{\pi _a})\) , the cone
\(W_{\pi _a} = B(I_{\pi _a})^0\)  is non-empty.
With Proposition REF  we now see that the
representation \((\pi _a, \Lambda (\mathcal {H}))\)  is semibounded.
To determine the cone \(W_{\pi _a}\) , we recall from Remark REF 
that for each \(x \in {\mathfrak {o}}_1(\mathcal {H})\) , the corresponding
operator \({\tt d}\pi _a(x)\)  on \(\Lambda (\mathcal {H})\)  is bounded. This means that
\({\mathfrak {o}}_1(\mathcal {H}) \subseteq B(I_{\pi _a}) \subseteq \overline{W_{\pi _a}}\) , and
since \(H(W_{\pi _a})\)  is a closed ideal (Lemma REF (v)),
it follows that
\( {\mathfrak {n}}:= {\mathfrak {z}}\oplus {\mathfrak {u}}_\infty (\mathcal {H}) \oplus {\mathfrak {p}}_2 \subseteq H(W_{\pi _a}). \) 
In view of \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H}) = {\mathfrak {u}}(\mathcal {H}) + {\mathfrak {n}}\) , this proves that
\(W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})} = C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {n}}\subseteq W_{\pi _a}= {\mathfrak {n}}+ (W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H})). \) 
It therefore remains to show that
\( W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H}) \subseteq C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {u}}_\infty (\mathcal {H}). \) 
To verify this assertion, let \(A = A^*\)  be a hermitian
operator on \(\mathcal {H}\)  and \({\tt d}\pi _a(A)\)  denote the corresponding operator
on \(\Lambda ^n(\mathcal {H})\) . Suppose that there exists an \(\varepsilon > 0\) 
such that the range of the spectral projection
\(P([\varepsilon ,\infty [)\)  is infinite dimensional. Then there exists
for each \(n\)  an orthonormal subset \(v_1, \ldots , v_n\)  in this space.
We then have
\(&\ \ \ \ \langle {\tt d}\pi _a(A)(v_1 \wedge \cdots \wedge v_n),v_1 \wedge \cdots \wedge v_n \rangle \\&= \sum _{j = 1}^n \langle v_1 \wedge \cdots \wedge Av_j \wedge \cdots \wedge v_n,v_1 \wedge \cdots \wedge v_n \rangle = \sum _{j = 1}^n \langle Av_j, v_j \rangle \ge n \varepsilon ,\) 

and this implies that \({\tt d}\pi _a(A)\)  is not bounded from above.
If, conversely, for every \(\varepsilon > 0\) 
the spectral projection \(P([\varepsilon ,\infty [)\)  has finite dimensional
range, then \(A = A_- + A_+\) , where
\(A_- \le 0\)  and \(A_+\)  is compact, i.e.,
\(iA \in \overline{C_{{\mathfrak {u}}(\mathcal {H})}} + {\mathfrak {u}}_\infty (\mathcal {H})\) .
This implies that the open cone \(W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H})\) 
is contained in \(C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {u}}_\infty (\mathcal {H})\) .
\(\Box \)
Remark 10.8  
(a) With a similar argument as in Remark REF  ones
argues that under the action of the identity component
\(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\)  on the space \(\Lambda (\mathcal {H})\) 
decomposes into two irreducible subrepresentations
\(\Lambda ^{\rm even}(\mathcal {H})\)  and \(\Lambda ^{\rm odd}(\mathcal {H})\) .
However, the action of the full group \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
is irreducible because the elements \(g \in \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
not contained in the identity component exchange
the two subspaces
\(\Lambda ^{\rm even}(\mathcal {H})\)  and
\(\Lambda ^{\rm odd}(\mathcal {H})\)  ([22]}).
(b) There is also an analog of the Banach–Lie algebra
\(\mathfrak {hsp}_{\rm res}(\mathcal {H})\)  acting irreducibly on \(\Lambda (\mathcal {H})\) 
in the fermionic case. Here it is an infinite dimensional
analog of the odd orthogonal Lie algebra \(\mathop {{\mathfrak {so} }}\nolimits _{2n+1}({\mathbb {R}})\) .
To see this Lie algebra, we define linear and antilinear
rank-one operators on \(\mathcal {H}\)  by
\( P_{v,w}(x) := \langle x,w \rangle v \quad \mbox{ and } \quad \overline{P}_{v,w}(x) := \langle w,x\rangle v \) 
and observe that \(P_{v,w}^* = P_{w,v}\)  and
\(\overline{P}_{v,w}^* = \overline{P}_{w,v}\) .
Therefore \(Q_{v,w} := P_{v,w} - P_{w,v} \in {\mathfrak {u}}_1(\mathcal {H})\) ,
and a direct calculation
yields
\( {\tt d}\pi _a(Q_{v,w}) = a_0^*(v)a_0(w) - a_0^*(w)a_0(v). \) 
As \(\mathop {{\rm tr}}\nolimits (Q_{v,w}) = 2i \mathop {{\rm Im}}\nolimits \langle v,w\rangle \) , we obtain with the map
\(\sigma \)  from (REF ) (Remark REF )
\( {\tt d}\pi _a(\sigma (Q_{v,w})) = {\tt d}\pi _a(-\mathop {{\rm Im}}\nolimits \langle v,w \rangle , Q_{v,w}). \) 
For the antilinear operators \(\overline{Q}_{v,w} :=\overline{P}_{v,w} - \overline{P}_{w,v}\)  we have
\(\overline{Q}_{v,w} \in {\mathfrak {o}}_1(\mathcal {H}) \subseteq {\mathfrak {o}}_{1,2}(\mathcal {H})\)  and
\( {\tt d}\pi _a(\overline{Q}_{v,w})= a_0^*(v)a_0^*(w) - a_0(w) a_0(v) = a_0^*(v)a_0^*(w) + a_0(v) a_0(w). \) 
Next we observe that the operators \(\rho (v) := \frac{1}{\sqrt{2}}(a_0(v) -a_0^*(v))\)  satisfy
\(&\ \ \ \ [\rho (v), \rho (w)]= {\textstyle {\frac{1}{2}}}[a_0(v) - a_0^*(v), a_0(w) - a_0^*(w)] \\&= (a_0(v)a_0(w) + a_0^*(v)a_0^*(w))+ (a_0^*(w) a_0(v) - a_0^*(v)a_0(w)) + i\mathop {{\rm Im}}\nolimits \langle v, w \rangle \\&= {\tt d}\pi _a(\mathop {{\rm Im}}\nolimits \langle v, w \rangle , \overline{Q}_{v,w}-Q_{v,w})={\tt d}\pi _a(\sigma (\overline{Q}_{v,w}- Q_{v,w})).\) 

This calculation implies that we obtain on the direct sum
\(\mathcal {H}\oplus {\mathfrak {o}}_{1,2}(\mathcal {H})\)  a Banach–Lie algebra structure
with the bracket
\( [(v,X), (v^{\prime },X^{\prime })]:= (Xv^{\prime } - X^{\prime }v, [X,X^{\prime }] + \overline{Q}_{v,w}-Q_{v,w}), \) 
and
\( (v,X) \mapsto \rho (v) + {\tt d}\pi _a(\sigma (X)) \) 
defines a representation by unbounded operators on \(\Lambda (\mathcal {H})\) ,
where the subalgebra \(\mathcal {H}\oplus {\mathfrak {o}}_1(\mathcal {H})\)  is represented by bounded
operators. A closer inspection shows that
\(\mathcal {H}\oplus {\mathfrak {o}}_1(\mathcal {H}) \cong {\mathfrak {o}}_1(\mathcal {H}\oplus {\mathbb {R}})\)  is an
infinite dimensional version of \({\mathfrak {o}}_{2n+1}({\mathbb {R}})\) 
(cf. [26]}).

Definition 10.9  Let \({\mathcal {I}}_\beta \subseteq \mathop {{\rm GL}}\nolimits (\mathcal {H}_{\mathbb {R}})\)  be the set of
orthogonal real-linear complex structures on the real
Hilbert space \(\mathcal {H}_{\mathbb {R}}\) . This set parameterizes the
complex Hilbert space structures on \(\mathcal {H}_{\mathbb {R}}\)  compatible
with the given real Hilbert space structure.

Lemma 10.10  The following assertions hold:
(i)
\({\mathcal {I}}_\beta = \mathop {\rm O{}}\nolimits (\mathcal {H}) \cap {\mathfrak {o}}(\mathcal {H})= \lbrace  g \in \mathop {{\rm GL}}\nolimits (\mathcal {H}_{\mathbb {R}}) \colon  g^\top = g^{-1} = -g \rbrace \)  is a submanifold
of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\) .
(ii)
The conjugation action of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  on
\({\mathcal {I}}_\beta \)  leads to a diffeomorphism \({\mathcal {I}}_\beta \cong \mathop {\rm O{}}\nolimits (\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) .
(iii)
\({\mathcal {I}}_\beta ^{\rm res} := \mathop {{\rm Ad}}\nolimits (\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}))I= \lbrace  J \in {\mathcal {I}}_\beta \colon  \Vert I - J \Vert _2 < \infty \rbrace \cong \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) .


Proof.   
(i) It only remains to show that \({\mathcal {I}}_\beta \)  is a
submanifold of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\) . For \(J \in {\mathcal {I}}_\beta \)  we parameterize
a neighborhood of \(J\)  by the map
\({\mathfrak {o}}(\mathcal {H}) \rightarrow \mathop {\rm O{}}\nolimits (\mathcal {H}),x \mapsto J e^x\) 
which is a diffeomorphism on some open 0-neighborhood \(U\subseteq {\mathfrak {o}}(\mathcal {H})\) , which we may assume to be invariant under
\(\mathop {{\rm Ad}}\nolimits (J)\) . That \(J e^x\)  is a complex structure is equivalent to
\( -Je^x = (Je^x)^{-1} = e^{-x} J^{-1} = - J e^{-J^{-1}xJ}, \) 
which is equivalent to
\(Jx = - xJ\) . We conclude that \({\mathcal {I}}_\beta \)  is a submanifold
of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  whose tangent space can be identified
with the set of \(J\) -antilinear elements in \({\mathfrak {o}}(\mathcal {H})\) .
(ii) Let \(I \in {\mathcal {I}}_\beta \)  be the canonical complex structure
given by \(Iv = iv\)  and \(J \in {\mathcal {I}}_\beta \) .
Then \(\mathcal {H}= (\mathcal {H}_{\mathbb {R}}, I)\)  and \((\mathcal {H}_{\mathbb {R}}, J)\)  are two complex Hilbert
spaces whose underlying real Hilbert spaces are isomorphic.
This implies that they have the same complex Hilbert dimension,
i.e., there exists a unitary isomorphism
\(g \colon  (\mathcal {H}_{\mathbb {R}}, I) \rightarrow (\mathcal {H}_{\mathbb {R}}, J)\) , i.e., \(g \in \mathop {\rm O{}}\nolimits (\mathcal {H})\)  with
\(gI = Jg\) . Therefore \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  acts transitively on
\({\mathcal {I}}_\beta \)  and the stabilizer of \(I\)  is the subgroup
\(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  (cf. [29]}).
Since its Lie algebra \({\mathfrak {u}}(\mathcal {H})\)  is complemented by
the closed subspace
\({\mathfrak {p}}:= \lbrace  x \in {\mathfrak {o}}(\mathcal {H}) \colon  Ix = - xI\rbrace ,\) 
it follows that \(\mathop {\rm O{}}\nolimits (\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\)  is a Banach homogeneous space
diffeomorphic to \({\mathcal {I}}_\beta \) .
(iii) For \(g = g_1 + g_2 \in \mathop {\rm O{}}\nolimits (\mathcal {H})\)  the operator
\([I,g] = [I, g_2] = 2 I g_2\)  is Hilbert–Schmidt if and only if
\(g_2\)  is, i.e., \(g \in \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\) .
This in turn is equivalent to
\(gIg^{-1} - I = [g,I]g^{-1}\)  being Hilbert–Schmidt, so that
\( \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}) = \lbrace  g \in \mathop {\rm O{}}\nolimits (\mathcal {H}) \colon \Vert gIg^{-1} - I \Vert _2 < \infty \rbrace  \) 
and therefore (ii) implies (iii).
\(\Box \)
Remark 10.11  A priori, the \(C^*\) -algebra
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  depends on the complex structure on \(\mathcal {H}\) , but
it can also be expressed as the \(C^*\) -algebra generated by
the hermitian elements \(b(f) := \frac{1}{\sqrt{2}}(a(f) + a^*(f))\)  satisfying
\( \lbrace b(f), b(g)\rbrace = {\textstyle {\frac{1}{2}}}\lbrace a(f), a^*(g)\rbrace  + {\textstyle {\frac{1}{2}}}\lbrace a^*(f), a(g)\rbrace = {\textstyle {\frac{1}{2}}}(\langle g, f \rangle + \langle f, g\rangle ) \mathbf {1} = \beta (f,g) \mathbf {1}. \) 
In this sense it is the \(C^*\) -envelope of the Clifford algebra
of \((\mathcal {H}_{\mathbb {R}},\beta )\) .
From this point of view it is even more transparent why the group
\(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  acts by automorphisms.
Now we can think of the Fock representation \(\pi _0\)  as
depending on the complex structure \(I\)  on \(\mathcal {H}\) , and any other
representation \(\pi _0 \circ \alpha _g\)  is the Fock representation
corresponding to the complex structure \(gIg^{-1} \in {\mathcal {I}}_\beta \) .
We thus obtain a map of \({\mathcal {I}}_\beta \)  into the set \(\mathop {{\rm Ext}}\nolimits (S(\mathop {{\rm CAR}}\nolimits (\mathcal {H})))\) 
of pure states of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) , mapping
\(J := gIg^{-1}\)  to \(\omega _J(A) := \langle \pi _0(\alpha _g^{-1}(A))\Omega ,\Omega \rangle \) . These states are called the Fock states of
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) , and in [29]} they are essentially
characterized as the pure quasi-free states (cf. Subsection REF  below).

Theorem 10.12   For the irreducible subrepresentation
\((\pi _a^+, \Lambda ^{\rm even}(\mathcal {H}))\)  of \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
the momentum set is the closed convex hull of the coadjoint orbit of
\(\lambda := \Phi _{\pi _a}([\Omega ])\) . This functional is given
by
\(\lambda (t,x) = t\quad \mbox{ on } \quad \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H}) = {\mathbb {R}}\oplus {\mathfrak {o}}_{\rm res}(\mathcal {H}). \) 


Proof.   
We want to apply Theorem REF .
For the subgroup \(K := \widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H})\subseteq G := \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\) 
with Lie algebra
\({\mathfrak {k}}= {\mathfrak {z}}\oplus {\mathfrak {u}}(\mathcal {H})\) , we have
\( {\mathfrak {g}}_{\mathfrak {p}}_+ \oplus {\mathfrak {k}}_{\mathfrak {p}}_-,\quad \mbox{ where } \quad {\mathfrak {p}}_\pm = \ker (\mathop {{\rm ad}}\nolimits I \mp 2i\mathbf {1}) \) 
and \(I \in {\mathfrak {u}}(\mathcal {H})\)  is the multiplication with \(i\)  on \(\mathcal {H}\) .
Since the Lie algebra \({\mathfrak {k}}\)  of \(K\)  is complemented by the closed
subspace \({\mathfrak {p}}_2\)  of antilinear elements in \({\mathfrak {g}}\) , the coset space
\(G/K\)  carries the structure of a Banach homogeneous space
(cf. Lemma REF ).
Moreover, the closed \(K\) -invariant subalgebra
\({\mathfrak {k}}_{\mathfrak {p}}_+\)  determines on \(G/K\)  the structure of a
complex manifold for which the tangent space in the base point
can be identified with \({\mathfrak {g}}_({\mathfrak {k}}_ {\mathfrak {p}}_+) \cong {\mathfrak {p}}_-\) 
([15]}).
In view of (REF ), we obtain with
[9]} that the
group \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  acts transitively on the homogeneous
space \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}_\beta _/P\) ,
where \(P\)  is the stabilizer of the subspace \(\mathcal {H}\oplus \lbrace 0\rbrace \subseteq \mathcal {H}_ (cf.\ Example~\ref {ex:liegrp}(d),(i)).From \) P Ores(H) = U(H)\( it now follows thatthe action of the Banach--Lie group \) G\( on \) G/K\( actually extends to aholomorphic action of a complex group.\) Let \(\Omega \in \Lambda ^0(\mathcal {H})\)  be the vacuum vector.
Then \(\Omega \)  is a smooth vector
which is an eigenvector for \(K\) . On \(\Lambda (\mathcal {H})\) , the operator
\(-i{\tt d}\pi (I)\)  is diagonal, and \(\Lambda ^n(\mathcal {H})\)  is the eigenspace
corresponding to the eigenvalue \(n\) . As \(-i\mathop {{\rm ad}}\nolimits I\)  acts on \({\mathfrak {p}}_-\)  with the
eigenvalue \(-2\) , it follows that
\({\tt d}\pi _a({\mathfrak {p}}_-)\Omega = \lbrace 0\rbrace ,\) 
so that Theorem REF  provides a holomorphic equivariant map
\( \eta \colon  G/K\rightarrow {\mathbb {P}}(\Lambda (\mathcal {H})^{\prime }), \quad g \mapsto \langle \cdot , \pi _a(g)^{-1} \Omega \rangle \) 
and a realization of \((\pi _a, \Lambda ^{\rm even}(\mathcal {H}))\) 
by holomorphic sections of a complex line bundle over \(G/K\) .
Since the representation
\((\pi _a, \Lambda ^{\rm even}(\mathcal {H}))\)  of \(G\) 
is irreducible, \(\Omega \)  is a cyclic vector.
As we have argued above, the action of every one-parameter subgroup of
\(G\)  extends to a holomorphic action of \( on \) G/K\(,so that Theorem~\ref {thm:5.7}(c) implies that\) I\( is the closed convex hull of the coadjoint orbitof \)\(. That \)\( has the desired form followsas in the proof of Theorem~\ref {thm:9.14}.\hfill \Box \)
Quasi-free representations
Let \(P = P^* = P^2\)  be an
orthogonal projection on \(\mathcal {H}\) ,
\(\mathcal {H}_- := \mathop {{\rm im}}\nolimits P\) , \(\mathcal {H}_+ := \ker P\) , and
\(\Gamma \colon  \mathcal {H}\rightarrow \mathcal {H}\)  be an isometric antilinear involution
commuting with \(P\) . Then
\(\tau _P := (\mathbf {1} - P) + \Gamma P \in \mathop {\rm O{}}\nolimits (\mathcal {H})\) .
We write \(a_0 \colon  \mathcal {H}\rightarrow B(\Lambda (\mathcal {H}))\)  for the map corresponding to
the Fock representation of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) 
(Remark REF ).
Twisting with the Bogoliubov automorphism defined by \(\tau _P\) , we
obtain an irreducible
representation \((\pi _P, \Lambda (\mathcal {H}))\)  of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  by
\( a_P(f) := a_0((\mathbf {1}-P)f) + a_0^*(\Gamma Pf)= \pi _0(a_{\tau _P}(f)) \quad \mbox{ for }\quad f \in \mathcal {H}, \) 
i.e., \(\pi _P = \pi _0 \circ \alpha _{\tau _P}\) .
These representations are called quasi-free.
For \(P =0\)  we recover the Fock representation defined by \(a_0\) .
Two quasi-free representations \(a_P\)  and \(a_Q\)  are equivalent if and only if
\(\Vert P - Q\Vert _2 < \infty \)  ([33]}).
Remark 10.13  The physical interpretation of \(P\)  is that its
range \(\mathcal {H}_-\)  consists
of the negative energy states and its kernel \(\mathcal {H}_+\)  consists of the
positive energy states.
For \(f \in \mathcal {H}_+\)  we have
\(a_P(f)\Omega = 0\)  and likewise
\(a_P^*(f)\Omega = 0\)  for \(f \in \mathcal {H}_-\) .
This is interpreted in such a way that the
creation operator \(a_P^*(f)\)  cannot create an additional negative
state from \(\Omega \)  because all negative states are already
filled (Pauli's Principle).
Likewise, the annihilation operator \(a_P(f)\) , corresponding
to the positive energy vector \(f\) , cannot extract any positive
energy state from \(\Omega \) .

The restricted unitary group \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
(cf. Example REF (c)) is a subgroup of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\) ,
and for all projections \(Q = gPg^{-1}\) , \(g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
we have \(\Vert P - Q\Vert _2 < \infty \) , so that the equivalence of
\(a_P\)  and \(a_Q\)  leads to a projective unitary representation
of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H})\)  on \(\Lambda (\mathcal {H})\)  determined by
\( \pi _a^P(g) a_P(f) \pi _a^P(g)^* = a_P(gf), \quad f \in \mathcal {H},g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H}). \) 
Let \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H})\)  denote the corresponding central
extension and write \(\pi _a^P\)  for its unitary representation
on \(\Lambda (\mathcal {H})\) . To see that we thus obtain a semibounded
representation of a Lie group, we first note that
we have an embedding
\(\iota \colon  \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \mathop {\rm O{}}\nolimits (\mathcal {H}), \quad g \mapsto \tau _P g \tau _P \) 
for which \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  is precisely the inverse image
of \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  and that for \(g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  we have
\(\pi _a(\tau _P g \tau _P)a_P(f)\pi _a(\tau _P g \tau _P)^*&= \pi _a(\tau _P g \tau _P)\pi _0(\alpha _{\tau _P} a(f))\pi _a(\tau _P g \tau _P)^*\\&= \pi _0(\alpha _{\tau _P g} a(f)) = \pi _P(a_g(f)) = a_P(gf),\) 

so that the projective representation of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  on
\(\Lambda (\mathcal {H})\)  coincide with \(\pi _a \circ \iota \) 
(cf. [1]}). It follows in particular that
\(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P) \cong \iota ^*\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
is a Lie group, \(\pi _a^P\)  is a smooth representation,
and since \(\mathop {{\rm im}}\nolimits (\mathop {\bf L{}}\nolimits (\iota ))\)  intersects the cone \(C_{{\mathfrak {u}}(\mathcal {H})}\subseteq W_{\pi _a}\) , the representation \(\pi _a^P\)  is semibounded
(cf. Proposition REF (iv)).
More precisely, each element \(x = (x_+, x_-) \in {\mathfrak {u}}(\mathcal {H}_+) \oplus {\mathfrak {u}}(\mathcal {H}_-)\)  with
\(i x_+ <\!\!< 0\)  and \(ix_- >\!\!> 0\)  lies in \(W_{\pi _a^P}\) .
This holds in particular for \(\tau _p I \tau _p = I(\mathbf {1} - 2P)\) .
In the physics literature, the
corresponding selfadjoint operator
\(Q := -i {\tt d}\pi _a^P(I) = {\tt d}\pi _a(\mathbf {1} - 2P)\) 
is called the charge operator and \(\mathbf {1} - 2P\)  the one-particle charge operator.
Its 1-eigenspace is \(\mathcal {H}_+\)  and its \(-1\) -eigenspace is \(\mathcal {H}_-\) .
Remark 10.14  The situation described above can be viewed as a
“second quantization” procedure that can be used to
turn a self-adjoint operator \(A\)  on the single particle space
\(\mathcal {H}\)  into a non-negative operator \(\widehat{A}\)  on the many particle
space. In fact, let \(P := P(]-\infty ,0[)\)  denote the
spectral projection of \(A\)  corresponding to the open negative axis.
Then \(iA\)  generates a strongly continuous one-parameter group
\(\gamma _A(t) := e^{itA}\) 
of \(\mathop {\rm U{}}\nolimits (\mathcal {H}_+) \times \mathop {\rm U{}}\nolimits (\mathcal {H}_-) \subseteq \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) ,
and \(\pi _a^P(\gamma _A(t)) = e^{it \widehat{A}}\)  is a one-parameter group
of \(\mathop {\rm U{}}\nolimits (\Lambda (\mathcal {H}))\)  whose infinitesimal generator
\(\widehat{A}\)  has non-negative spectrum.

For more details on the complex manifolds
\(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\)  (the isotropic restricted Graßmannian)
and \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)/(\mathop {\rm U{}}\nolimits (P(\mathcal {H})) \times \mathop {\rm U{}}\nolimits (\mathcal {H}_+))\) 
(the restricted or Sato–Segal–Wilson Graßmannian) we refer to
[22]} and [36]}, where one also finds a detailed
discussion of the corresponding complex line bundles,
the Pfaffian line bundle \({\rm Pf}\)  over \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) 
whose dual permits the even spin representation as a space of
holomorphic sections, and the determinant line bundle \({\rm Det}\) 
over the restricted Graßmannian whose dual provides a
realization of the representations of \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P)\) 
mentioned above. Physical aspects of highest weight representations
of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H})\)  are discussed in [37]}.

Perspectives
Classification problems:
In the preceding sections we mainly discussed three
prototypical classes of semibounded representations:
highest weight representations of the Virasoro group,
the metaplectic representation on the bosonic Fock space
and the spin representation on the fermionic Fock space.
These representations are of fundamental importance in
mathematical physics and homomorphisms from a Lie group
\(G\)  to \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) , \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  or
\(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
can be used to obtain semibounded representations of a central
extension by pulling back the representations discussed above.
This construction has been a major source of (projective) representations
for loop groups \(G = C^\infty ({\mathbb {S}}^1,K)\)  and \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)\) 
(cf. [38]}, [22]}, [40]}, [37]}).
What still remains to be developed is a better global perspective
on semibounded representations, including classification
results on irreducible ones and the existence of direct integral
decompositions.
Problem 11.1  Classify all irreducible semibounded unitary representations
of the groups \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) , \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
and \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P)\) , where \(\mathcal {H}\)  is a complex Hilbert space and
\(P\)  an orthogonal projection on \(\mathcal {H}\) .
These classification problems are special cases of the more general
problem of the classification of the projective semibounded
unitary representations of the automorphism groups
of hermitian Hilbert symmetric spaces. For the three groups above,
the corresponding spaces of
\({\mathcal {I}}_\omega ^{\rm res}\) , \({\mathcal {I}}_\beta ^{\rm res}\) , resp.,
the restricted Graßmannian \(\mathop {{\rm Gr}}\nolimits _{\rm res}(\mathcal {H},P) :=\lbrace  gPg^{-1} \colon  g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\rbrace \) . A crucial
difference between these spaces is that the first one
is equivalent to a symmetric Hilbert domain
(of negative curvature) and the latter two are positively
curved spaces. This difference is also reflected in the difference
between the cones \(W_{\pi _s}\)  and \(W_{\pi _s}\) 
in \(\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\)  and \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) 
(Theorems REF  and REF ).
An even larger class of groups arises as automorphism groups
of Hilbert flag manifolds such as the orbit of a finite
flag in \(\mathcal {H}\)  under the group \(\mathop {\rm U{}}\nolimits _2(\mathcal {H})\) . For a systematic
discussion of these manifolds and the topology of the
corresponding real and complex groups we refer to
[9]} and for corresponding representations
to [43]}.

For loop groups of the form \(C^\infty ({\mathbb {S}}^1,K)\) ,
\(K\)  a compact connected Lie group, the irreducible
projective positive energy representations can be identified
as highest weight representations ([22]},
[45]}, and in particular
[38]} for \(K = ).With the convexity theorems in \cite {AP83} and \cite {KP84} it shouldbe possible to show that these representationsare semibounded and one can hope for an analog ofTheorem~\ref {thm:8.22} asserting that every irreduciblesemibounded representation of the corresponding double extensioneither is a highest weight representation or its dual.In \cite {Ne09b} it is shown on the algebraic level that therealso exist many interesting unitary representationsof infinite rank generalizations of twisted and untwistedloop algebras, resp., their double extensions(so-called locally affine Lie algebras). On the group levelthey correspond to (double extensions of) groupsof the form \) C(S1, K)\(, where\) K\( is a Hilbert--Lie group, such as \) U2(H)\(.It seems quite likely that all these representationsare semibounded with momentum sets generated by a singlecoadjoint orbit.\)New sources of semibounded representations:
Although many interesting classes of semibounded representations
are known, a systematic understanding of the geometric sources
of these representations is still lacking.
As we have seen above, on the Lie algebra level the existence
of open invariant cones in a trivial central extension
is necessary (Remark REF ). However, as the example
\(\mathcal {V}({\mathbb {S}}^1)\)  shows, it is not sufficient
(Theorem REF ).
Clearly, the circle \({\mathbb {S}}^1\)  plays a special role in many constructions,
as the rich theory for central extensions of loop groups and the
Virasoro group shows. Beyond \({\mathbb {S}}^1\) , it seems that
Lie algebras of conformal vector fields
(as generalizations of \(\mathcal {V}({\mathbb {S}}^1)\) ) (cf. Example REF , [47]} and [48]}) and Lie algebras of
sections of vector bundles over Lorentzian manifolds
(or more general “causal” spaces, [49]})
are natural candidates to be investigated
with respect to the existence of semibounded representations.
The latter class of Lie algebras is a natural
generalization of loop algebras.
Here an interesting point is that, although the conformal
groups \(\mathop {\rm Conf{}}\nolimits ({\mathbb {S}}^1) = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)\)  of the circle and the conformal
group \(\mathop {\rm Conf{}}\nolimits ({\mathbb {S}}^{1,1})\)  of the Lorentzian torus
are infinite dimensional ,
for Lorentzian manifolds of dimension \(\ge 3\)  the conformal groups are finite
dimensional (cf. [50]}). In particular, it is contained in the list
of [47]}. This leads to the well-studied class of hermitian
Lie groups (see in particular [48]}, [53]}, [49]}).
Coadjoint orbits: There is a symplectic version of semibounded
representations, namely Hamiltonian actions
\(\sigma \colon  G \times M \rightarrow M\)  with a momentum map
\(\Phi \colon  M \rightarrow {\mathfrak {g}}^{\prime }\)  for which the image of the momentum map
is semi-equicontinuous. If \(G\)  is finite dimensional
and \(\Phi (M)\)  is closed, then Proposition REF 
implies that there exists an \(x \in {\mathfrak {g}}\)  for which the
Hamiltonian function \(H_x(m) := \Phi (m)(x)\)  is proper.
In particular, \(\Phi \)  is a proper map. In this sense
the semi-equicontinuity of the image of \(\Phi \)  is
a weakening of properness, which is a useful property
as far as convexity properties
are concerned ([55]}).
Even though we do not know in general to which extent coadjoint
orbits of infinite dimensional Lie groups are manifolds, the orbit
\(\mathcal {O}_\lambda \)  can always be viewed as the range of a momentum map
\(\Phi \colon  G \rightarrow {\mathfrak {g}}^{\prime }, g \mapsto \mathop {{\rm Ad}}\nolimits ^*(g)\lambda \)  corresponding to the
left action of \(G\)  on itself preserving the left invariant
closed 2-form \(\Omega \)  with \(\Omega _1(x,y) = \lambda ([x,y])\) .
This action is semibounded if and only if \(\mathcal {O}_\lambda \)  is
semi-equicontinuous.
Since the momentum sets of semibounded representations always consists
of semi-equicontinuous orbits, the identification of the set
\({\mathfrak {g}}^{\prime }_{\rm seq}\)  of semi-equicontinuous coadjoint orbits of a given
Lie algebra is already a solid first step towards the understanding
of corresponding semibounded representations.
As we have seen in many situations above, a useful tool to
study convexity properties of coadjoint orbits
are projection maps
\(p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {t}}\) , where \({\mathfrak {t}}\subseteq {\mathfrak {g}}\)  is a
“compactly embedded” subalgebra, i.e.,
the action of \(e^{\mathop {{\rm ad}}\nolimits {\mathfrak {t}}}\)  on \({\mathfrak {g}}\)  factor through
the action of a compact abelian group.
As we have seen in Section , sometimes one
does not want to project to abelian subalgebras and
one has to study projections \(p_{\mathfrak {k}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {k}}\) ,
where \({\mathfrak {k}}\)  is a subalgebra for which \(e^{\mathop {{\rm ad}}\nolimits {\mathfrak {k}}}\) 
leaves a norm on \({\mathfrak {g}}\)  invariant
This is of particular interest to understand open invariant
cones \(W \subseteq {\mathfrak {g}}\)  because they often have the form
\( W = \mathop {{\rm Ad}}\nolimits (G) (W \cap {\mathfrak {k}}) \quad \mbox{ with } \quad p_{\mathfrak {k}}(W) = W \cap {\mathfrak {k}}\) 
(cf. Theorem REF ).
In this situation on needs convexity theorems of the type
\( p_{\mathfrak {k}}(\mathcal {O}_x) \subseteq \overline{\mathop {{\rm conv}}\nolimits }(\mathop {{\rm Ad}}\nolimits (N_G({\mathfrak {k}}))x) + C, \) 
where \(C\)  is a certain invariant convex cone in \({\mathfrak {k}}\)  and
\(N_G({\mathfrak {k}}) \subseteq G\)  is the normalizer of \({\mathfrak {k}}\)  in \(G\) 
(cf. [56]} for finite dimensional Lie algebras).
If \({\mathfrak {k}}= {\mathfrak {t}}\)  is abelian, then \(N_G({\mathfrak {t}})\)  is an analog
of the Weyl group.
For infinite dimensional Lie algebras, not many convexity theorems
are known. Of relevance for semibounded representations
is the particular case of affine Kac–Moody Lie algebras
([57]}, [58]}), and for Lie algebras of bounded
operators on Hilbert spaces
the infinite dimensional version of Kostant's Theorem by
A. Neumann ([59]}) is crucial. What is still lacking
is a uniform framework for results of this type.

Smooth vectors for representations
Let \(G\)  be a Lie group with Lie algebra \({\mathfrak {g}}\)  and exponential
function \(\exp _G \colon  {\mathfrak {g}}\rightarrow G\) .
Further, let \(V\)  be a locally convex space and
\(\pi \colon  G \rightarrow \mathop {{\rm GL}}\nolimits (V)\)  be a homomorphism
defining a continuous action of \(G\)  on \(V\) . We write
\(\pi ^v(g) := \pi (g)v\)  for the orbit maps and
\(V^\infty := \lbrace  v \in V \colon  \pi ^v \in C^\infty (G,V) \rbrace  \) 
for the space of smooth vectors.
In this appendix we collect some results of
[60]} that are used in the present paper.
Let
\({\tt d}\pi \colon  {\mathfrak {g}}\rightarrow \mathop {{\rm End}}\nolimits (V^\infty ), \quad {\tt d}\pi (x)v := \frac{d}{dt} \hbox{\vrule width0.5ptheight 5mm depth 3mm${{}\atop {{}\atop {\scriptstyle t=0}}}$} \pi (\exp tx)v \) 
denote the derived action of \({\mathfrak {g}}\)  on \(V^\infty \) .
That this is indeed a representation of \({\mathfrak {g}}\)  follows by observing that
the map \(V^\infty \rightarrow C^\infty (G,V), v \mapsto \pi ^v\) 
intertwines the action of \(G\)  with the right translation action on
\(C^\infty (G,V)\) , and this implies that the derived action
corresponds to the action of \({\mathfrak {g}}\)  on \(C^\infty (G,V)\)  by left invariant
vector fields (cf. [45]} for details).
Definition A.1  Let \(G\)  be a Banach–Lie group and
write \(\mathcal {P}(V)\)  for the set of continuous seminorms on \(V\) .
For each \(p \in \mathcal {P}(V)\)  and \(n \in {\mathbb {N}}_0\)  we define a seminorm
\(p_n\)  on \(V^\infty \)  by
\( p_n(v) := \sup \lbrace  p({\tt d}\pi (x_1) \cdots {\tt d}\pi (x_n)v) \colon \Vert x_i\Vert \le 1\rbrace  \) 
and endow \(V^\infty \)  with the locally convex topology defined by these
seminorms.

Theorem A.2  
If \((\pi , V)\)  is a representation of the Banach–Lie group
\(G\)  on the locally convex space \(V\)  defining a continuous action
of \(G\)  on \(V\) , then the
action \(\sigma (g,v) := \pi (g)v\)  of \(G\)  on \(V^\infty \)  is smooth.
If \(V\)  is a Banach space, then \(V^\infty \)  is complete,
i.e., a Fréchet space.

Theorem A.3  
If \((\pi , \mathcal {H})\)  is a unitary representation of a Lie group
\(G\) , then \(v \in \mathcal {H}\)  is a smooth vector if and only if the
corresponding matrix coefficient \(\pi ^{v,v}(g) := \langle \pi (g)v,v\rangle \) 
is smooth on a \(\mathbf {1}\) -neighborhood in \(G\) . If, in addition,
\(v\)  is cyclic, then the representation is smooth.

In the following we write \(\overline{u}\)  for the image of a unitary
operator \(u \in \mathop {\rm U{}}\nolimits (\mathcal {H})\)  in the projective unitary group
\(\mathop {\rm PU{}}\nolimits (\mathcal {H}) :=\mathop {\rm U{}}\nolimits (\mathcal {H})/\) .
Theorem A.4   Let \(G\)  be a connected Lie group,
\(\mathcal {H}\)  a complex Hilbert space and
\(\pi \colon  G \rightarrow \mathop {\rm U{}}\nolimits (\mathcal {H})\)  be a map with \(\pi (\mathbf {1}) = \mathbf {1}\) 
for which the corresponding map
\(\overline{\pi }\colon  G \rightarrow \mathop {\rm PU{}}\nolimits (\mathcal {H})\)  is a group homomorphism.
If there exists a \(v \in \mathcal {H}\)  for which
the function \((g,h) \mapsto \langle \pi (g)\pi (h)v, v\rangle \)  is
smooth on a neighborhood of \((\mathbf {1},\mathbf {1})\)  in \(G \times G\) ,
then the central
extension
\(\widehat{G} := \overline{\pi }^*\mathop {\rm U{}}\nolimits (\mathcal {H}) = \lbrace  (g,u) \in G \times \mathop {\rm U{}}\nolimits (\mathcal {H})\colon  \overline{\pi }(g) = \overline{u} \rbrace  \) 
of \(G\)  by \( is a Lie group and\) v\( is a smooth vector for therepresentation \) (, H)\( of \) G\( by\) (g,u) := u\(.\)
Proof.   
To exhibit \(\widehat{G}\)  as a Lie group, we have to show that
there exists a section \(\sigma \colon  G \rightarrow \widehat{G}\)  for which the
corresponding 2-cocycle
\( f_\sigma (g_1, g_2)= \sigma (g_1) \sigma (g_2) \sigma (g_1 g_2)^{-1} \) 
is smooth in a neighborhood of
\((\mathbf {1},\mathbf {1})\)  ([62]}).
Here we use that \(G\)  is connected.
A particular section \(\sigma \colon  G \rightarrow \widehat{G}\)  is given by
\(\sigma (g) = (g,\pi (g))\) .
Let \(U \subseteq G\)  be an open \(\mathbf {1}\) -neighborhood such that
\( \langle \pi (g)v,\pi (h)^{-1}v \rangle \ne 0 \quad \mbox{ for } \quad g,h \in U. \) 
Its existence follows from our continuity assumption.
If \(U^{\prime }\subseteq U\)  is an open \(\mathbf {1}\) -neighborhood with \(U^{\prime }U^{\prime } \subseteq U\) , we then
have for \(g_1, g_2 \in U^{\prime }\)
\( f_\sigma (g_1, g_2) \langle \pi (g_1 g_2)v, v\rangle = \langle \pi (g_1) \pi (g_2) v, v\rangle , \) 
which leads to
\( f_\sigma (g_1, g_2) = \frac{\langle \pi (g_1)\pi (g_2) v, v\rangle }{\langle \pi (g_1 g_2)v, v\rangle }. \) 
Therefore \(f_\sigma \)  is smooth in a neighborhood of \((\mathbf {1},\mathbf {1})\) .
This shows that \(\widehat{G}\)  is a Lie group and the multiplication map
\(G \times \widehat{G}, (g,t) \mapsto (g, t\pi (g))\) 
is smooth in a neighborhood of \(\mathbf {1}\)  ([62]}).
The representation
\(\widehat{\pi }\)  now satisfies
\( \langle \widehat{\pi }(g,t\pi (g))v, v \rangle = \langle t\pi (g)v, v \rangle = t \langle \pi (g)v,v\rangle , \) 
which is smooth in a neighborhood of \(\mathbf {1}\) . Now
Theorem REF  implies that \(v\)  is a smooth
vector for the representation \(\widehat{\pi }\) .
\(\Box \)
Remark A.5  The assumption that \(G\)  is connected in the preceding theorem
can be removed if \(G\)  is a Banach–Lie group. In this case we assume,
in addition, that \(\overline{\pi }\)  is continuous in the sense that
all functions \(g \mapsto |\langle \pi (g)v,w \rangle |\)  on \(G\)  are continuous
(cf. [64]}).
Then \(\widehat{G}\)  is the pullback of a central extension
of topological groups \(\mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \mathop {\rm PU{}}\nolimits (\mathcal {H})\) , hence in particular
a topological group with respect to the topology inherited from
\(G \times \mathop {\rm U{}}\nolimits (\mathcal {H})\) . Under the assumptions of Theorem REF ,
the central extension \(\widehat{G}_0\)  of the identity component
\(G_0\)  of \(G\)  carries a natural Lie group structure compatible
with the given topology on \(\widehat{G}\) . For each \(g \in \widehat{G}\) , the conjugation
map \(c_g\)  induces a continuous automorphism of the Banach–Lie group
\(\widehat{G}_0\) , and since continuous homomorphisms of Banach–Lie groups
are automatically smooth ([65]}),
\(c_g\)  also defines a Lie automorphism of
\(\widehat{G}_0\) . This implies that \(\widehat{G}\)  carries a unique Lie group
structure which coincides on the open subgroup \(\widehat{G}_0\)  with the given
one (see also [62]}).

Remark A.6  
In the situation of Theorem REF ,
the Lie algebra cocycle defining the central extension
\(\widehat{{\mathfrak {g}}}= \mathop {\bf L{}}\nolimits (\widehat{G})\)  of \({\mathfrak {g}}\)  by \({\mathbb {R}}\)  in the sense that
\( \widehat{{\mathfrak {g}}}= {\mathbb {R}}\oplus _{\eta } {\mathfrak {g}}, \quad [(t,x), (t^{\prime },x^{\prime })] = (\eta (x,x^{\prime }), [x,x^{\prime }]) \) 
can be calculated as follows.

If \({\tt d}\pi \colon  {\mathfrak {g}}\rightarrow \mathop {{\rm End}}\nolimits (\mathcal {H}^\infty )\)  is the map obtained
from the representation \({\tt d}\widehat{\pi }\)  via
\({\tt d}\pi (x) := {\tt d}\widehat{\pi }(0,x)\) , then we have for each unit
vector \(v \in \mathcal {H}^\infty \)  the relation
\( [{\tt d}\pi (x), {\tt d}\pi (y)]v= {\tt d}\pi ([x,y])v + i \eta (x,y)v, \) 
so that
\(\eta (x,y)&= \mathop {{\rm Im}}\nolimits \langle [{\tt d}\pi (x), {\tt d}\pi (y)]v, v\rangle - \mathop {{\rm Im}}\nolimits \langle {\tt d}\pi ([x,y])v, v \rangle \\&= 2\mathop {{\rm Im}}\nolimits \langle {\tt d}\pi (x)v, {\tt d}\pi (y) v\rangle + i \langle {\tt d}\pi ([x,y])v, v \rangle .\) 

The cone of positive definite forms on a Banach space
Let \(V\)  be a Banach space and
\(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  be the Banach space of continuous symmetric bilinear
maps \(\beta \colon  V \times V \rightarrow {\mathbb {R}}\) , endowed with the norm
\(\Vert \beta \Vert  := \sup \lbrace  |\beta (v,v)| \colon  \Vert v\Vert  \le 1\rbrace . \) 
Clearly, the set \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  of positive semidefinite
bilinear maps is a closed convex cone in \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) .
Lemma B.1 

The cone \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  has interior points if and only if
\(V\)  is topologically isomorphic to a Hilbert space. If this is
the case, then its interior consists of all those positive definite
forms \(\beta \)  for which the norm \(\Vert v\Vert _\beta := \sqrt{\beta (v,v)}\) 
is equivalent to the norm on \(V\) .

Proof.   
Suppose first that \(V\)  carries a Hilbert space
structure \(\beta \) . Replacing the original norm by an equivalent
Hilbert norm, we may assume that \(V\)  is a real Hilbert space.
Then \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  can be identified with the space
\(\mathop {{\rm Sym}}\nolimits (V)\)  of symmetric operators on \(V\)  by assigning to
\(A \in \mathop {{\rm Sym}}\nolimits (V)\)  the form \(\beta _A(v,v) = \langle Av,w\rangle \) , satisfying
\( |\beta _A(v,v)| = |(Av,v)| \le \Vert A\Vert \Vert v\Vert ^2,\quad \mbox{ so that } \quad \Vert \beta _A\Vert  \le \Vert A\Vert .\) 
The polarization identity
\( (Av,w) = \beta _A(v,w) = \frac{1}{4} (\beta _A(v+w,v+w) -\beta _A(v-w,v-w)) \) 
further implies that \(\Vert A\Vert  \le 2 \beta _A\) , so that the
Banach spaces \(\mathop {{\rm Sym}}\nolimits (V)\)  and \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  are topologically isomorphic.
The identity \(\mathop {{\rm id}}\nolimits _V = \mathbf {1}\)  is an interior point in the cone
of positive operators, \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  has interior points.
Suppose, conversely, that \(\beta \in \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  is an
interior point. Then \(\beta - \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  is a 0-neighborhood
in \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) , which implies the existence of some
\(c > 0\)  such that \(\Vert \gamma \Vert  \le c\)  implies
\(\gamma (v,v) \le \beta (v,v)\)  for all \(v \in V\) .
Fixing \(v_0 \in V\) , we pick \(\alpha \in V^{\prime }\)  with
\(\Vert \alpha \Vert  = 1\)  and \(\alpha (v_0) = \Vert v_0\Vert \) .
Then \(\gamma (v,w) := c\alpha (v)\alpha (w)\)  is a symmetric bilinear form
with \(\Vert \gamma \Vert  = c\Vert \alpha \Vert ^2 = c\) . We therefore obtain
\( c \Vert v_0\Vert ^2 = \gamma (v_0, v_0) \le \beta (v_0, v_0)\le \Vert \beta \Vert  \Vert v_0\Vert ^2, \) 
showing that \(\beta \)  is positive definite and that the norm
\(\sqrt{\beta (v,v)}\)  is equivalent to the norm on \(V\) .
\(\Box \)
Remark B.2  
Let \((V,\omega )\)  be a weakly symplectic Banach space, i.e.,
\(\omega \)  is a non-degenerate skew-symmetric bilinear form.
Then
\( \mathfrak {sp}(V,\omega ) := \lbrace  X \in B(V) \colon  (\forall v,w\in V)\,\omega (Xv,w) + \omega (v,Xw) = 0\rbrace  \) 
is the corresponding symplectic Lie algebra. In particular,
\(X \in B(V)\)  belongs to \(\mathfrak {sp}(V,\omega )\)  if and only if
the bilinear form \(\omega (Xv,w)\)  is symmetric. The corresponding
quadratic function \(H_X(v) := {\textstyle {\frac{1}{2}}}\omega (Xv,v)\)  is called the
Hamiltonian function defined by \(X\).
If there exists an \(X \in \mathfrak {sp}(V,\omega )\)  for which
\((v,w) := \omega (Xv,w)\)  defines a Hilbert space structure
on \(V\) , then each continuous linear
functional \(\alpha \in V^{\prime }\)  is of the form \(i_v\omega \)  for
some \(v \in V\) . This means that
\((V,\omega )\)  is strongly symplectic,
i.e., the map
\(\Phi _\omega \colon  V \rightarrow V^{\prime }, v \mapsto i_v\omega \)  is surjective,
hence a topological isomorphism by the Open Mapping Theorem.
We further see that \(X\)  is injective. To see that it is also surjective,
let \(u \in V\)  and represent the continuous linear functional
\(i_u\omega \)  as \((w,\cdot )\)  for some \(w \in V\) . Then
\(\omega (u,v) = (w,v) = \omega (Xw,v)\)  for all \(v \in V\) ,
and thus \(Xw = u\) . Hence \(X\)  is a topological isomorphism,
satisfying \(\omega (u,v) = (X^{-1}u,v)\)  for \(v,w \in V\) . This further implies
that \(X\)  is skew-symmetric.

Proposition B.3  
Let \((V,\omega )\)  be a strongly symplectic Banach space.
Then the convex cone \(\lbrace  X \in \mathfrak {sp}(V,\omega ) \colon  H_X \ge 0\rbrace \)  has interior
points if and only if \(V\)  is topologically isomorphic to a
Hilbert space. If this is the case,
then there exists a complex structure \(I \in \mathfrak {sp}(V,\omega )\)  for
which \(H_I\)  defines a compatible Hilbert space structure on \(V\) , i.e.,
\( \omega (v,w) = \mathop {{\rm Im}}\nolimits \langle v, w\rangle \) 
for the underlying complex Hilbert space.

Proof.   
(cf. [67]})
If \(\beta \colon  V \times V \rightarrow {\mathbb {R}}\)  is a symmetric bilinear form,
then \(\Phi _\beta \colon  V \rightarrow V^{\prime }, v \mapsto i_v\beta \)  also is a continuous
linear map, and
\(B := \Phi _\omega ^{-1} \circ \Phi _\beta \colon  V \rightarrow V\)  is continuous
linear with
\( \omega (Bv,v) = \Phi _\omega (Bv)(v)= \Phi _\beta (v)(v) = \beta (v,v), \) 
so that every symmetric bilinear form can be represented by an
element of \(\mathfrak {sp}(V,\omega )\) , and we obtain a topological isomorphism
\(\mathfrak {sp}(V,\omega ) \cong \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) . Combining this with
Lemma REF  proves the first assertion.
We now assume that \(V\)  is a real Hilbert space with the scalar product
\((v,w) = \omega (Xv,w)\) , where \(X\)  is as above. Then \(A := X^{-1}\)  is an
invertible skew-symmetric operator with \(\omega (v,w) = (Av,w)\) .
Then the complex linear extension \(A_ to the complex Hilbertspace \) V yields a self-adjoint operator \(iA_.The complex conjugation \)\( of \) V with respect to \(V\) 
now satisfies \(\sigma \circ iA_\sigma = - i A_, sothat \) Spec(iA\( is a compact symmetric subset of \) R\(, notcontaining \) 0\(. We therefore have an orthogonal decomposition\) V V+ V-\( into the positive and negative spectralsubspaces of \) iA. Since \(\sigma (V_\pm ) = V_\mp \) , we obtain
an isomorphism \(V \cong V_+\) , and hence a complex structure
\(I\)  on \(V\) , corresponding to multiplication by \(i\)  on \(V_+\) .
This means that \(v_+ \in V_+\)  with \(v = v_+ + \sigma (v_+)\) 
satisfies
\(\omega (Iv,v)&= (AIv,v)= (AI(v_+ + \sigma (v_+)), v_+ + \sigma (v_+)) \\&= (A_i v_+ - i \sigma (v_+)), v_+ + \sigma (v_+))= i (A_v_+ - \sigma (v_+)), v_+ + \sigma (v_+)) \\&= i \big ((A_v̏_+, v_+) - (v_+, A_v̏_+)\big )= 2 (iA_v̏_+, v_+),\) 

so we obtain a complex structure \(I\)  on \(V\) 
for which the Hamiltonian \(H_I\)  defines a Hilbert structure on
\(V\) . Since \(I\)  is skew-symmetric, \((V,I)\)  inherits the structure
of a complex Hilbert space with respect to the scalar product
\( \langle v, w \rangle := \omega (Iv,w) + i \omega (Iv,Iw) = \omega (Iv,w) + i \omega (v,w). \) 
Therefore our assumption leads to the representation
of \(\omega \)  as \(\omega (v,w) = \mathop {{\rm Im}}\nolimits \langle v, w \rangle \)  for a complex
Hilbert space structure on \(V\) .
\(\Box \)
Involutive Lie algebras with root decomposition
Definition C.1   (a) We call an abelian subalgebra
\({\mathfrak {t}}\)  of the real Lie algebra \({\mathfrak {g}}\)  a compactly embedded
Cartan subalgebra if \({\mathfrak {t}}\)  is maximal abelian and \(\mathop {{\rm ad}}\nolimits {\mathfrak {t}}\) 
is simultaneously diagonalizable on the complexification \({\mathfrak {g}}_with purely imaginary eigenvalues.Then we have a {\it root decomposition}\)\( {\mathfrak {g}}_ {\mathfrak {t}}_ \sum _{\alpha \in \Delta } {\mathfrak {g}}_\alpha , \)\(where \) g= { x g (h t [h,x]= (h)
x}\( and\)\(\Delta := \lbrace  \alpha \in {\mathfrak {t}}_* \setminus \lbrace 0\rbrace  \colon  {\mathfrak {g}}_\alpha \ne  \lbrace 0\rbrace \rbrace \)\( is the corresponding {\it root system}.\) If \(\sigma \colon  {\mathfrak {g}}_{\mathfrak {g}}_ denotes the complex conjugation withrespect to \) g\(, we write \) x* := -(x)\( for \) x g, so that
\({\mathfrak {g}}= \lbrace  x \in {\mathfrak {g}}_ x^* = -x\rbrace \) .
We then have
(I1)
\(\alpha (x) \in {\mathbb {R}}\)  for \(x \in i {\mathfrak {t}}\) .
(I2)
\(\sigma ({\mathfrak {g}}_\alpha ) = {\mathfrak {g}}_{-\alpha }\)  for \(\alpha \in \Delta .\)


Lemma C.2 
 For \(0 \ne  x_{\alpha } \in {\mathfrak {g}}_{\alpha }\)  the subalgebra
\({\mathfrak {g}}_x_\alpha ) := \mathop {{\rm span}}\nolimits \lbrace x_\alpha , x_{\alpha }^*,[x_\alpha , x_{\alpha }^*]\rbrace \) 
is \(\sigma \) -invariant and of one of the following types:
(A)
The abelian type: \([x_\alpha , x_{\alpha }^*] = 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha )\)  is two dimensional abelian.
(N)
The nilpotent type: \([x_\alpha , x_{\alpha }^*] \ne  0\) 
and \(\alpha ([x_\alpha , x_{\alpha }^*]) = 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha )\)  is a three dimensional Heisenberg algebra.
(S)
The simple type: \(\alpha ([x_\alpha , x_{\alpha }^*]) \ne  0\) ,
i.e., \({\mathfrak {g}}_x_\alpha ) \cong \mathop {{\mathfrak {sl} }}\nolimits _2(\) . In this case we distinguish the
two cases:
(CS)
\(\alpha ([x_\alpha , x_{\alpha }^*]) > 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\cong \mathop {{\mathfrak {su} }}\nolimits _2(\) , and
(NS)
\(\alpha ([x_\alpha , x_{\alpha }^*]) < 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\cong \mathop {{\mathfrak {su} }}\nolimits _{1,1}( \cong \mathop {{\mathfrak {sl} }}\nolimits _2({\mathbb {R}})\) .


Proof.   
First we note that, in view of \(x_\alpha ^* \in {\mathfrak {g}}_{-\alpha }\) ,
[26]} applies, and we see that \({\mathfrak {g}}_x_\alpha )\)  is of one of
the three types (A), (N) and (S). We note that
\(\alpha ([x_\alpha , x_\alpha ^*]) \in {\mathbb {R}}\) 
because of (I2) and \([x_\alpha , x_\alpha ^*] \in i{\mathfrak {t}}\) .
Now it is easy to check that
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\)  is of type (CS), resp., (NS), according to the sign
of this number.
\(\Box \)
The following proposition provides useful information for the
analysis of invariant cones and orbit projections.
Here we write \(p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {t}}\)  for the
projection along \([{\mathfrak {t}},{\mathfrak {g}}] = \sum _{\alpha } ({\mathfrak {g}}_\alpha + {\mathfrak {g}}_{-\alpha }) \cap {\mathfrak {g}}\) ,
and \(p_{{\mathfrak {t}}^*} \colon  {\mathfrak {g}}^* \rightarrow {\mathfrak {t}}^*\)  for the restriction map.
Proposition C.3 

Let \(x \in {\mathfrak {t}}\) , \(x_\alpha \in {\mathfrak {g}}_\alpha \)  and
\(\lambda \in {\mathfrak {t}}^*\) . Then the following assertions hold:
(i)
\(p_{\mathfrak {t}}(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits (x_\alpha - x_\alpha ^*)}x)= x +{\left\lbrace \begin{array}{ll}{\mathbb {R}}^+ \alpha (x) [x_\alpha ^*, x_\alpha ] & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) \le 0 \\[0,2] { \frac{\alpha (x)}{\alpha ([x_\alpha , x_\alpha ^*])}}[x_\alpha ^*, x_\alpha ] & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) > 0.\end{array}\right.}\)
(ii)
\(p_{{\mathfrak {t}}^*}(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits ^* (x_\alpha - x_\alpha ^*)}\lambda )= \lambda +{\left\lbrace \begin{array}{ll}{\mathbb {R}}^+ \lambda ([x_\alpha ^*, x_\alpha ]) \alpha & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) \le 0 \\[0,2] \frac{\lambda ([x_\alpha ^*, x_\alpha ])}{\alpha ([x_\alpha , x_\alpha ^*])}\alpha & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) > 0.\end{array}\right.}\)


Proof.   
(i) is an immediate consequence of [56]},
and (ii) follows from (i) and the relation
\(p_{{\mathfrak {t}}^*}(\lambda )(e^{\mathop {{\rm ad}}\nolimits y}x)= \lambda (p_{\mathfrak {t}}(e^{\mathop {{\rm ad}}\nolimits y}x)).\) 
\(\Box \)
Some facts on Fock spaces
Let \(\mathcal {H}\)  be a complex Hilbert space. We endow the \(n\) -fold tensor
product with the canonical Hilbert structure defined by
\( \langle v_1 \otimes \cdots \otimes v_n, w_1 \otimes \cdots \otimes w_n \rangle := \prod _{j = 1}^n \langle v_j, w_j \rangle , \) 
and form the Hilbert space direct sum
\(T(\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} \mathcal {H}^{\otimes n}\) .
In \(\mathcal {H}^{\otimes n}\)  we write \(S^n(\mathcal {H})\)  for the closed subspace
generated by the symmetric tensors and
\(\Lambda ^n(\mathcal {H})\)  for the closed subspace generated by the
alternating tensors. We thus obtain subspaces
\( S(\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} S^n(\mathcal {H})\quad \mbox{ and } \quad \Lambda (\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} \Lambda ^n(\mathcal {H}) \) 
of \(T(\mathcal {H})\)  and write \(P_s\) , resp., \(P_a\)  for the corresponding
orthogonal projections:
\( P_s(f_1 \otimes \cdots \otimes f_n)= \frac{1}{n!} \sum _{\sigma \in S_n} f_{\sigma (1)} \otimes \cdots \otimes f_{\sigma (n)} \) 
and
\( P_a(f_1 \otimes \cdots \otimes f_n)= \frac{1}{n!} \sum _{\sigma \in S_n} \mathop {{\rm sgn}}\nolimits (\sigma )f_{\sigma (1)} \otimes \cdots \otimes f_{\sigma (n)}. \) 
The dense subspace \(S(\mathcal {H})_0 := \sum _{n \ge 0} S^n(\mathcal {H})\)  of \(S(\mathcal {H})\) 
carries an associative
algebra structure, given by
\( f_1 \vee \cdots \vee f_n := \sqrt{n!}P_s(f_1 \otimes \cdots \otimes f_n) \) 
and likewise \(\Lambda (\mathcal {H})_0 := \sum _{n \ge 0} \Lambda ^n(\mathcal {H})\) 
inherits an algebra structure defined by
\( f_1 \wedge \cdots \wedge f_n := \sqrt{n!}P_a(f_1 \otimes \cdots \otimes f_n). \) 
A unit vector \(\Omega \)  in the one-dimensional space
\(S^0(\mathcal {H}) = \Lambda ^0(\mathcal {H})\)  is called a vacuum vector.
Lemma D.1 
 We have
\(T \vee S= \sqrt{{n+m \atopwithdelims ()n}} P_s(T \otimes S) \quad \mbox{ for } \quad T \in S^n(\mathcal {H}), S \in S^m(\mathcal {H})\) 
and
\(T \wedge S= \sqrt{{n+m \atopwithdelims ()n}} P_a(T \otimes S) \quad \mbox{ for } \quad T \in \Lambda ^n(\mathcal {H}), S \in \Lambda ^m(\mathcal {H}).\) 


Proof.   
For the symmetric case we first note that
\(f^n = \sqrt{n!} f^{\otimes n}\) , so that
\( f^n \vee g^m= \sqrt{(n+m)!} P_s(f^{\otimes n} \otimes g^{\otimes m})= \sqrt{{n+m \atopwithdelims ()n}} P_s(f^n \otimes g^m). \) 
Since the elements \(f^n\)  generate \(S^n(\mathcal {H})\)  topologically,
(REF ) follows.
For the alternating case we obtain for
\(T = f_1 \wedge \cdots \wedge f_n\)  and
\(S = g_1 \wedge \cdots \wedge g_m\)  the relation
\(&\ \ \ \ \frac{1}{\sqrt{(n+m)!}}T \wedge S= P_a(f_1 \otimes \cdots \otimes g_m)\\&= P_a(P_a(f_1 \otimes \cdots \otimes f_n) \otimes P_a(g_1 \otimes \cdots \otimes g_m))= \frac{1}{\sqrt{n!m!}} P_a(T \otimes S).\) 

\(\Box \)
Remark D.2  
(a) For the norms of the product of \(T \in S^n(\mathcal {H})\)  and \(S \in S^m(\mathcal {H})\) ,
we obtain with (REF )
\( \frac{1}{\sqrt{(n+m)!}} \Vert T \vee S\Vert = \frac{1}{\sqrt{n!}\sqrt{m!}} \Vert P_s(T \otimes S)\Vert \le \frac{1}{\sqrt{n!}\sqrt{m!}} \Vert T \otimes S\Vert = \frac{\Vert T\Vert }{\sqrt{n!}} \frac{\Vert S\Vert }{\sqrt{m!}}.\) 
From that we derive for \(T \in S^k(\mathcal {H})\)  and \(n \in {\mathbb {N}}\)  the relation
\( \frac{1}{\sqrt{(kn)!}} \Vert T^n\Vert \le \Big (\frac{1}{\sqrt{k!}}\Vert T\Vert \Big )^n, \) 
and for \(T \in S^2(\mathcal {H})\)  we find in particular
\(\Vert e^T\Vert ^2= \sum _{n = 0}^\infty \frac{1}{(n!)^2} \Vert T^n\Vert ^2\le \sum _{n = 0}^\infty \frac{(2n)!}{(n!)^2 2^n} \Vert T\Vert ^{2n},\) 
which converges for \(\Vert T\Vert ^2 < \frac{1}{2}\) .
(b) For \(T \in \Lambda ^n(\mathcal {H})\)  and \(S \in \Lambda ^m(\mathcal {H})\) 
we likewise obtain with (REF )
\( \frac{1}{\sqrt{(n+m)!}} \Vert T \wedge S\Vert \le \frac{\Vert T\Vert }{\sqrt{n!}} \frac{\Vert S\Vert }{\sqrt{m!}},\) 
which leads for \(T \in \Lambda ^2(\mathcal {H})\)  to
\(\Vert e^T\Vert ^2= \sum _{n = 0}^\infty \frac{1}{(n!)^2} \Vert T^n\Vert ^2\le \sum _{n = 0}^\infty \frac{(2n)!}{(n!)^2 2^n} \Vert T\Vert ^{2n}.\) 

Lemma D.3  
Let \(A\)  be an antilinear Hilbert–Schmidt operator on \(\mathcal {H}\) 
and define \(A^*\)  by \(\langle A^*v,w \rangle = \langle Aw,v\rangle \) .
If \(A^* = A\) , then there exists a unique element
\(\widehat{A} \in S^2(\mathcal {H})\)  with
\( \langle \widehat{A}, f_1 \vee f_2 \rangle = \langle Af_1, f_2 \rangle f\quad \mbox{ for } \quad f_1, f_2 \in \mathcal {H}, \) 
and if \(A^* = - A\) , there exists a unique element \(\widehat{A} \in \Lambda ^2(\mathcal {H})\) 
with
\( \langle \widehat{A}, f_1 \wedge f_2 \rangle = \langle Af_1, f_2 \rangle \quad \mbox{ for } \quad f_1, f_2 \in \mathcal {H}. \) 
Moreover, the following assertions hold:
(i)
\(\Vert \widehat{A}\Vert ^2 = {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2\) .
(ii)
If \(A\)  and \(B\)  are hermitian and antilinear, then
\(\langle \widehat{A}, \widehat{B} \rangle = {\textstyle {\frac{1}{2}}}\mathop {{\rm tr}}\nolimits (AB)\) .
(iii)
If \(A\)  and \(B\)  are skew-hermitian and antilinear, then
\(\langle \widehat{A}, \widehat{B} \rangle = -{\textstyle {\frac{1}{2}}}\mathop {{\rm tr}}\nolimits (AB)\) .


Proof.   
(a) First we consider the case where \(A^* = A\) .
Let \((e_j)_{j \in J}\)  be an orthonormal basis of \(\mathcal {H}\) .
Then we have in \(S^2(\mathcal {H})\)  the relations
\( \Vert e_j^2\Vert ^2 = 2\Vert  e_j \otimes e_j\Vert ^2 = 2\quad \mbox{ and } \quad \Vert e_j \vee e_k\Vert ^2= 2\Vert {\textstyle {\frac{1}{2}}}(e_j \otimes e_k + e_k \otimes e_j)\Vert ^2 = 1, \quad j\ne k. \) 
If \(<\)  denotes a linear order on \(J\) , we thus obtain the
orthonormal basis
\(\frac{1}{\sqrt{2}} e_j^2, \quad e_j \vee e_k, \quad j < k, \quad \mbox{ of } \quad S^2(\mathcal {H}).\) 
This leads to
\(\Vert \widehat{A}\Vert ^2&= \sum _j {\textstyle {\frac{1}{2}}}|\langle \widehat{A}, e_j^2 \rangle |^2+ \sum _{j<k} |\langle \widehat{A}, e_j \vee e_k \rangle |^2 \\&= \sum _j {\textstyle {\frac{1}{2}}}|\langle Ae_j, e_j \rangle |^2+ \sum _{j<k} |\langle Ae_j, e_k \rangle |^2= {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2.\) 
For \(A^* = -A\)  we similarly get
\(\Vert \widehat{A}\Vert ^2&= \sum _{j<k} |\langle \widehat{A}, e_j \vee e_k \rangle |^2= \sum _{j<k} |\langle Ae_j, e_k \rangle |^2 = {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2.\) 
(b) Let \(\mathop {{\rm Herm}}\nolimits _2(\mathcal {H})_a\)  denote the complex subspace
of hermitian antilinear
Hilbert–Schmidt operators.
Then the prescription \(\langle A, B \rangle := \mathop {{\rm tr}}\nolimits (AB) = \mathop {{\rm tr}}\nolimits (AB^*)\)  defines a
sesquilinear form on this space with
\( \overline{\mathop {{\rm tr}}\nolimits (AB)} = \mathop {{\rm tr}}\nolimits ((AB)^*) = \mathop {{\rm tr}}\nolimits (B^*A^*) = \mathop {{\rm tr}}\nolimits (BA), \) 
so that it is hermitian. Its restriction to the diagonal
satisfies
\(\langle A,A \rangle = \mathop {{\rm tr}}\nolimits (A^2)= \sum _{j \in J} \langle A^2e_j, e_j \rangle = \sum _{j \in J} \langle Ae_j, Ae_j \rangle = \Vert A\Vert _2^2. \) 
In view of (a), polarization implies that
\(\mathop {{\rm tr}}\nolimits (AB) = \langle A, B \rangle = 2 \langle \widehat{A}, \widehat{B}\rangle \) 
for \(A, B \in \mathop {{\rm Herm}}\nolimits _2(\mathcal {H})_a\) .
(c) For the space \(\mathop {{\rm Aherm}}\nolimits _2(\mathcal {H})_a\)  of skew-hermitian antilinear
Hilbert–Schmidt operators the same argument works with
\(\langle A, B \rangle := -\mathop {{\rm tr}}\nolimits (AB) = \mathop {{\rm tr}}\nolimits (AB^*)\) .
\(\Box \)
